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Mathematics · Textbook solutions

Trigonometry - I

Every solved example, exercise, and miscellaneous question, in the order the textbook teaches them. · 123 questions

2.1 Trigonometric Functions

9 q

Solved Examples

Worked · 9
  1. 2.1 SolvedEx.1
    Find the signs of the following : i) sin⁡300∘\sin 300^\circ ii) cos⁡400∘\cos 400^\circ iii) cot⁡(−206∘)\cot(-206^\circ)
  2. 2.1 SolvedEx.2
    For θ=30∘\theta = 30^\circ, verify that sin⁡2θ=2sin⁡θcos⁡θ\sin 2\theta = 2\sin\theta\cos\theta.
  3. 2.1 SolvedEx.3
    Evaluate the following : i) cos⁡30∘×cos⁡60∘+sin⁡30∘×sin⁡60∘\cos 30^\circ \times \cos 60^\circ + \sin 30^\circ \times \sin 60^\circ ii) 4cos⁡345∘−3cos⁡45∘+sin⁡45∘4\cos^3 45^\circ - 3\cos 45^\circ + \sin 45^\circ iii) sin⁡20+sin⁡2π6+sin⁡2π3+sin⁡2π2\sin^2 0 + \sin^2\frac{\pi}{6} + \sin^2\frac{\pi}{3} + \sin^2\frac{\pi}{2} iv) sin⁡π+2cos⁡π+3sin⁡3π2+4cos⁡3π2−5sec⁡π−6cosec⁡3π2\sin\pi + 2\cos\pi + 3\sin\frac{3\pi}{2} + 4\cos\frac{3\pi}{2} - 5\sec\pi - 6\operatorname{cosec}\frac{3\pi}{2}
  4. 2.1 SolvedEx.4
    Find all trigonometric functions of the angle made by OP with X-axis where P is (−5,12)(-5, 12).
  5. 2.1 SolvedEx.5
    sec⁡θ=−3\sec\theta = -3 and π<θ<3π2\pi < \theta < \frac{3\pi}{2} then find the values of other trigonometric functions.
  6. 2.1 SolvedEx.6
    If sec⁡x=135\sec x = \frac{13}{5}, xx lies in the fourth quadrant, find the values of other trigonometric functions.
  7. 2.1 SolvedEx.7
    If tan⁡A=43\tan A = \frac{4}{3}, find the value of 2sin⁡A−3cos⁡A2sin⁡A+3cos⁡A\frac{2\sin A - 3\cos A}{2\sin A + 3\cos A}.
  8. 2.1 SolvedEx.8
    If sec⁡θ=2\sec\theta = \sqrt{2}, 3π2<θ<2π\frac{3\pi}{2} < \theta < 2\pi then find the value of 1+tan⁡θ+cosec⁡θ1+cot⁡θ−cosec⁡θ\frac{1 + \tan\theta + \operatorname{cosec}\theta}{1 + \cot\theta - \operatorname{cosec}\theta}.
  9. 2.1 SolvedEx.9
    If sin⁡θ=−35\sin\theta = -\frac{3}{5} and 180∘<θ<270∘180^\circ < \theta < 270^\circ then find all trigonometric functions of θ\theta.

Exercise 2.1

18 q
  1. Ex 2.1 Q1
    Find the trigonometric functions of 0∘0^\circ, 30∘30^\circ, 45∘45^\circ, 60∘60^\circ, 150∘150^\circ, 180∘180^\circ, 210∘210^\circ, 300∘300^\circ, 330∘330^\circ, −30∘-30^\circ, −45∘-45^\circ, −60∘-60^\circ, −90∘-90^\circ, −120∘-120^\circ, −225∘-225^\circ, −240∘-240^\circ, −270∘-270^\circ, −315∘-315^\circ
  2. State the signs of
    Ex 2.1 Q2(i)
    tan⁡380∘\tan 380^\circ
  3. Ex 2.1 Q2(ii)
    cot⁡230∘\cot 230^\circ
  4. Ex 2.1 Q2(iii)
    sec⁡468∘\sec 468^\circ
  5. Ex 2.1 Q3
    State the signs of cos⁡4c\cos 4^{c} and cos⁡4∘\cos 4^\circ. Which of these two is greater ?
  6. State the quadrant in which θ\theta lies if
    Ex 2.1 Q4(i)
    sin⁡θ<0\sin\theta < 0 and tan⁡θ>0\tan\theta > 0
  7. Ex 2.1 Q4(ii)
    cos⁡θ<0\cos\theta < 0 and tan⁡θ>0\tan\theta > 0
  8. Evaluate each of the following :
    Ex 2.1 Q5(i)
    sin⁡30∘+cos⁡45∘+tan⁡180∘\sin 30^\circ + \cos 45^\circ + \tan 180^\circ
  9. Ex 2.1 Q5(ii)
    cosec⁡45∘+cot⁡45∘+tan⁡0∘\operatorname{cosec} 45^\circ + \cot 45^\circ + \tan 0^\circ
  10. Ex 2.1 Q5(iii)
    sin⁡30∘×cos⁡45∘×tan⁡360∘\sin 30^\circ \times \cos 45^\circ \times \tan 360^\circ
  11. Ex 2.1 Q6
    Find all trigonometric functions of angle in standard position whose terminal arm passes through point (3,−4)(3, -4).
  12. Ex 2.1 Q7
    If cos⁡θ=1213\cos\theta = \frac{12}{13}, 0<θ<π20 < \theta < \frac{\pi}{2}, find the value of sin⁡2θ−cos⁡2θ2sin⁡θcos⁡θ\frac{\sin^2\theta - \cos^2\theta}{2\sin\theta\cos\theta}, 1tan⁡2θ\frac{1}{\tan^2\theta}
  13. Using tables evaluate the following :
    Ex 2.1 Q8(i)
    4cot⁡45∘−sec⁡260∘+sin⁡30∘4\cot 45^\circ - \sec^2 60^\circ + \sin 30^\circ
  14. Ex 2.1 Q8(ii)
    cos⁡20+cos⁡2π6+cos⁡2π3+cos⁡2π2\cos^2 0 + \cos^2\frac{\pi}{6} + \cos^2\frac{\pi}{3} + \cos^2\frac{\pi}{2}
  15. Find the other trigonometric functions if
    Ex 2.1 Q9(i)
    If cos⁡θ=−35\cos\theta = -\frac{3}{5} and 180∘<θ<270∘180^\circ < \theta < 270^\circ.
  16. Ex 2.1 Q9(ii)
    If sec⁡A=−257\sec A = -\frac{25}{7} and AA lies in the second quadrant.
  17. Ex 2.1 Q9(iii)
    If cot⁡x=34\cot x = \frac{3}{4}, xx lies in the third quadrant.
  18. Ex 2.1 Q9(iv)
    tan⁡x=−512\tan x = \frac{-5}{12}, xx lies in the fourth quadrant.

2.2 Fundamental Identities

19 q

Solved Examples

Worked · 19
  1. 2.2 SolvedEx.1
    Find the value of sin⁡41π4\sin\frac{41\pi}{4}.
  2. 2.2 SolvedEx.2
    Find the value of cos⁡765∘\cos 765^\circ.
  3. 2.2 SolvedEx.3
    If tan⁡θ+1tan⁡θ=2\tan\theta + \frac{1}{\tan\theta} = 2 then find the value of tan⁡2θ+1tan⁡2θ\tan^2\theta + \frac{1}{\tan^2\theta}.
  4. Which of the following is true?
    2.2 SolvedEx.4(i)
    2cos⁡2θ=1−tan⁡2θ1+tan⁡2θ2\cos^2\theta = \frac{1-\tan^2\theta}{1+\tan^2\theta}
  5. 2.2 SolvedEx.4(ii)
    cot⁡A−tan⁡Bcot⁡B−tan⁡A=cot⁡Atan⁡B\frac{\cot A - \tan B}{\cot B - \tan A} = \cot A \tan B
  6. 2.2 SolvedEx.4(iii)
    cos⁡θ1−tan⁡θ+sin⁡θ1−cot⁡θ=sin⁡θ+cos⁡θ\frac{\cos\theta}{1-\tan\theta} + \frac{\sin\theta}{1-\cot\theta} = \sin\theta + \cos\theta
  7. 2.2 SolvedEx.5
    If 5tan⁡A=25\tan A = \sqrt{2}, π<A<3π2\pi < A < \frac{3\pi}{2} and sec⁡B=11\sec B = \sqrt{11}, 3π2<B<2π\frac{3\pi}{2} < B < 2\pi then find the value of cosec⁡A−tan⁡B\operatorname{cosec} A - \tan B.
  8. 2.2 SolvedEx.6
    If tan⁡θ=17\tan\theta = \frac{1}{\sqrt{7}} then evaluate cosec⁡2θ−sec⁡2θcosec⁡2θ+sec⁡2θ\frac{\operatorname{cosec}^2\theta - \sec^2\theta}{\operatorname{cosec}^2\theta + \sec^2\theta}.
  9. 2.2 SolvedEx.7
    Prove that cos⁡6θ+sin⁡6θ=1−3sin⁡2θ cos⁡2θ\cos^6\theta + \sin^6\theta = 1 - 3\sin^2\theta\,\cos^2\theta.
  10. Eliminate θ\theta from the following :
    2.2 SolvedEx.8(i)
    x=acos⁡θx = a\cos\theta, y=bsin⁡θy = b\sin\theta
  11. 2.2 SolvedEx.8(ii)
    x=acos⁡3θx = a\cos^3\theta, y=bsin⁡3θy = b\sin^3\theta
  12. 2.2 SolvedEx.8(iii)
    x=2+3cos⁡θx = 2 + 3\cos\theta, y=5+3sin⁡θy = 5 + 3\sin\theta
  13. 2.2 SolvedEx.9
    If 2sin⁡2θ+7cos⁡θ=52\sin^2\theta + 7\cos\theta = 5 then find the permissible values of cos⁡θ\cos\theta.
  14. 2.2 SolvedEx.10
    Solve for θ\theta, if 4sin⁡2θ−2(3+1)sin⁡θ+3=04\sin^2\theta - 2(\sqrt{3}+1)\sin\theta + \sqrt{3} = 0.
  15. 2.2 SolvedEx.11
    If tan⁡θ+sec⁡θ=1.5\tan\theta + \sec\theta = 1.5 then find tan⁡θ\tan\theta, sin⁡θ\sin\theta and sec⁡θ\sec\theta.
  16. 2.2 SolvedEx.12
    Prove that sin⁡θ1−cos⁡θ+tan⁡θ1+cos⁡θ=sec⁡θ cosec⁡θ+cot⁡θ\frac{\sin\theta}{1-\cos\theta} + \frac{\tan\theta}{1+\cos\theta} = \sec\theta\,\operatorname{cosec}\theta + \cot\theta.
  17. 2.2 SolvedEx.13
    Prove that sec⁡θ−tan⁡θsec⁡θ+tan⁡θ=1−2sec⁡θtan⁡θ+2tan⁡2θ\frac{\sec\theta - \tan\theta}{\sec\theta + \tan\theta} = 1 - 2\sec\theta\tan\theta + 2\tan^2\theta.
  18. 2.2 SolvedEx.14
    Prove that (sec⁡A−tan⁡A)2=1−sin⁡A1+sin⁡A(\sec A - \tan A)^2 = \frac{1-\sin A}{1+\sin A}.
  19. 2.2 SolvedEx.15
    Find the polar co-ordinates of the point whose Cartesian coordinates are (3,3)(3, 3).

Exercise 2.2

36 q
  1. Ex 2.2 Q1
    If 2sin⁡A=1=2cos⁡B2\sin A = 1 = \sqrt{2}\cos B and π2<A<π\frac{\pi}{2} < A < \pi, 3π2<B<2π\frac{3\pi}{2} < B < 2\pi, then find the value of tan⁡A+tan⁡Bcos⁡A−cos⁡B\frac{\tan A + \tan B}{\cos A - \cos B}.
  2. Ex 2.2 Q2
    If sin⁡A3=sin⁡B4=15\frac{\sin A}{3} = \frac{\sin B}{4} = \frac{1}{5} and AA, BB are angles in the second quadrant then prove that 4cos⁡A+3cos⁡B=−54\cos A + 3\cos B = -5.
  3. Ex 2.2 Q3
    If tan⁡θ=12\tan\theta = \frac{1}{2}, evaluate 2sin⁡θ+3cos⁡θ4cos⁡θ+3sin⁡θ\frac{2\sin\theta + 3\cos\theta}{4\cos\theta + 3\sin\theta}.
  4. Eliminate θ\theta from the following :
    Ex 2.2 Q4(i)
    x=3sec⁡θx = 3\sec\theta, y=4tan⁡θy = 4\tan\theta
  5. Ex 2.2 Q4(ii)
    x=6cosec⁡θx = 6\operatorname{cosec}\theta, y=8cot⁡θy = 8\cot\theta
  6. Ex 2.2 Q4(iii)
    x=4cos⁡θ−5sin⁡θx = 4\cos\theta - 5\sin\theta, y=4sin⁡θ+5cos⁡θy = 4\sin\theta + 5\cos\theta
  7. Ex 2.2 Q4(iv)
    x=5+6cosec⁡θx = 5 + 6\operatorname{cosec}\theta, y=3+8cot⁡θy = 3 + 8\cot\theta
  8. Ex 2.2 Q4(v)
    2x=3−4tan⁡θ2x = 3 - 4\tan\theta, 3y=5+3sec⁡θ3y = 5 + 3\sec\theta
  9. Ex 2.2 Q5
    If 2sin⁡2θ+3sin⁡θ=02\sin^2\theta + 3\sin\theta = 0, find the permissible values of cos⁡θ\cos\theta.
  10. Ex 2.2 Q6
    If 2cos⁡2θ−11cos⁡θ+5=02\cos^2\theta - 11\cos\theta + 5 = 0 then find possible values of cos⁡θ\cos\theta.
  11. Ex 2.2 Q7
    Find the acute angle θ\theta such that 2cos⁡2θ=3sin⁡θ2\cos^2\theta = 3\sin\theta.
  12. Ex 2.2 Q8
    Find the acute angle θ\theta such that 5tan⁡2θ+3=9sec⁡θ5\tan^2\theta + 3 = 9\sec\theta.
  13. Ex 2.2 Q9
    Find sin⁡θ\sin\theta such that 3cos⁡θ+4sin⁡θ=43\cos\theta + 4\sin\theta = 4.
  14. Ex 2.2 Q10
    If cosec⁡θ+cot⁡θ=5\operatorname{cosec}\theta + \cot\theta = 5, then evaluate sec⁡θ\sec\theta.
  15. Ex 2.2 Q11
    If cot⁡θ=34\cot\theta = \frac{3}{4} and π<θ<3π4\pi < \theta < \frac{3\pi}{4} then find the value of 4cosec⁡θ+5cos⁡θ4\operatorname{cosec}\theta + 5\cos\theta.
  16. Find the Cartesian co-ordinates of points whose polar coordinates are :
    Ex 2.2 Q12(i)
    (3,90∘)(3, 90^\circ)
  17. Ex 2.2 Q12(ii)
    (1,180∘)(1, 180^\circ)
  18. Find the polar co-ordinates of points whose Cartesian co-ordinates are :
    Ex 2.2 Q13(i)
    (5,5)(5, 5)
  19. Ex 2.2 Q13(ii)
    (1,3)(1, \sqrt{3})
  20. Ex 2.2 Q13(iii)
    (−1,−1)(-1, -1)
  21. Ex 2.2 Q13(iv)
    (−3,1)(-\sqrt{3}, 1)
  22. Find the value of
    Ex 2.2 Q14(i)
    sin⁡19πc3\sin\frac{19\pi^c}{3}
  23. Ex 2.2 Q14(ii)
    cos⁡1140∘\cos 1140^\circ
  24. Ex 2.2 Q14(iii)
    cot⁡25πc3\cot\frac{25\pi^c}{3}
  25. Prove the following identities:
    Ex 2.2 Q15(i)
    (1+tan⁡2A)+(1+1tan⁡2A)=1sin⁡2A−sin⁡4A(1 + \tan^2 A) + \left(1 + \frac{1}{\tan^2 A}\right) = \frac{1}{\sin^2 A - \sin^4 A}
  26. Ex 2.2 Q15(ii)
    (cos⁡2A−1)(cot⁡2A+1)=−1(\cos^2 A - 1)(\cot^2 A + 1) = -1
  27. Ex 2.2 Q15(iii)
    (sin⁡θ+sec⁡θ)2+(cos⁡θ+cosec⁡θ)2=(1+cosec⁡θ sec⁡θ)2(\sin\theta + \sec\theta)^2 + (\cos\theta + \operatorname{cosec}\theta)^2 = (1 + \operatorname{cosec}\theta\,\sec\theta)^2
  28. Ex 2.2 Q15(iv)
    (1+cot⁡θ−cosec⁡θ)(1+tan⁡θ+sec⁡θ)=2(1 + \cot\theta - \operatorname{cosec}\theta)(1 + \tan\theta + \sec\theta) = 2
  29. Ex 2.2 Q15(v)
    tan⁡3θ1+tan⁡2θ+cot⁡3θ1+cot⁡2θ=sec⁡θ cosec⁡θ−2sin⁡θcos⁡θ\frac{\tan^3\theta}{1 + \tan^2\theta} + \frac{\cot^3\theta}{1 + \cot^2\theta} = \sec\theta\,\operatorname{cosec}\theta - 2\sin\theta\cos\theta
  30. Ex 2.2 Q15(vi)
    1sec⁡θ+tan⁡θ−1cos⁡θ=1cos⁡θ−1sec⁡θ−tan⁡θ\frac{1}{\sec\theta + \tan\theta} - \frac{1}{\cos\theta} = \frac{1}{\cos\theta} - \frac{1}{\sec\theta - \tan\theta}
  31. Ex 2.2 Q15(vii)
    sin⁡θ1+cos⁡θ+1+cos⁡θsin⁡θ=2cosec⁡θ\frac{\sin\theta}{1 + \cos\theta} + \frac{1 + \cos\theta}{\sin\theta} = 2\operatorname{cosec}\theta
  32. Ex 2.2 Q15(viii)
    tan⁡θsec⁡θ−1=sec⁡θ+1tan⁡θ\frac{\tan\theta}{\sec\theta - 1} = \frac{\sec\theta + 1}{\tan\theta}
  33. Ex 2.2 Q15(ix)
    cot⁡θcosec⁡θ−1=cosec⁡θ+1cot⁡θ\frac{\cot\theta}{\operatorname{cosec}\theta - 1} = \frac{\operatorname{cosec}\theta + 1}{\cot\theta}
  34. Ex 2.2 Q15(x)
    (sec⁡A+cos⁡A)(sec⁡A−cos⁡A)=tan⁡2A+sin⁡2A(\sec A + \cos A)(\sec A - \cos A) = \tan^2 A + \sin^2 A
  35. Ex 2.2 Q15(xi)
    1+3cosec⁡2θ⋅cot⁡2θ+cot⁡6θ=cosec⁡6θ1 + 3\operatorname{cosec}^2\theta \cdot \cot^2\theta + \cot^6\theta = \operatorname{cosec}^6\theta
  36. Ex 2.2 Q15(xii)
    1−sec⁡θ+tan⁡θ1+sec⁡θ−tan⁡θ=sec⁡θ+tan⁡θ−1sec⁡θ+tan⁡θ+1\frac{1 - \sec\theta + \tan\theta}{1 + \sec\theta - \tan\theta} = \frac{\sec\theta + \tan\theta - 1}{\sec\theta + \tan\theta + 1}

Miscellaneous Exercise 2

41 q

(I) Select the correct option

Practice · 10
  1. Misc I Q1
    The value of the expression cos⁡1∘⋅cos⁡2∘⋅cos⁡3∘⋅…⋅cos⁡179∘=\cos 1^\circ \cdot \cos 2^\circ \cdot \cos 3^\circ \cdot \ldots \cdot \cos 179^\circ =

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  2. Misc I Q2
    tan⁡A1+sec⁡A+1+sec⁡Atan⁡A\frac{\tan A}{1+\sec A}+\frac{1+\sec A}{\tan A} is equal to

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  3. Misc I Q3
    If α\alpha is a root of 25cos⁡2θ+5cos⁡θ−12=025\cos^2\theta + 5\cos\theta - 12 = 0, π2<α<π\frac{\pi}{2} < \alpha < \pi then sin⁡2α\sin 2\alpha is equal to :

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  4. Misc I Q4
    If θ=60∘\theta = 60^\circ, then 1+tan⁡2θ2tan⁡θ\frac{1+\tan^2\theta}{2\tan\theta} is equal to

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  5. Misc I Q5
    If sec⁡θ=m\sec\theta = m and tan⁡θ=n\tan\theta = n, then 1m{(m+n)+1(m+n)}\frac{1}{m}\left\{(m+n)+\frac{1}{(m+n)}\right\} is equal to

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  6. Misc I Q6
    If cosec⁡θ+cot⁡θ=52\operatorname{cosec}\theta + \cot\theta = \frac{5}{2}, then the value of tan⁡θ\tan\theta is

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  7. Misc I Q7
    1−sin⁡2θ1+cos⁡θ+1+cos⁡θsin⁡θ−sin⁡θ1−cos⁡θ1-\frac{\sin^2\theta}{1+\cos\theta}+\frac{1+\cos\theta}{\sin\theta}-\frac{\sin\theta}{1-\cos\theta} equals

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  8. Misc I Q8
    If cosec⁡θ−cot⁡θ=q\operatorname{cosec}\theta - \cot\theta = q, then the value of cot⁡θ\cot\theta is

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  9. Misc I Q9
    The cotangent of the angles π3\frac{\pi}{3}, π4\frac{\pi}{4} and π6\frac{\pi}{6} are in

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  10. Misc I Q10
    The value of tan⁡1∘ tan⁡2∘ tan⁡3∘…tan⁡89∘\tan 1^\circ\, \tan 2^\circ\, \tan 3^\circ \ldots \tan 89^\circ is equal to

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(II)

Practice · 31
  1. Misc II Q1
    Find the trigonometric functions of : 90∘90^\circ, 120∘120^\circ, 225∘225^\circ, 240∘240^\circ, 270∘270^\circ, 315∘315^\circ, −120∘-120^\circ, −150∘-150^\circ, −180∘-180^\circ, −210∘-210^\circ, −300∘-300^\circ, −330∘-330^\circ
  2. State the signs of
    Misc II Q2(i)
    cosec⁡520∘\operatorname{cosec} 520^\circ
  3. Misc II Q2(ii)
    cot⁡1899∘\cot 1899^\circ
  4. Misc II Q2(iii)
    sin⁡986∘\sin 986^\circ
  5. State the quadrant in which θ\theta lies if
    Misc II Q3(i)
    tan⁡θ<0\tan\theta < 0 and sec⁡θ>0\sec\theta > 0
  6. Misc II Q3(ii)
    sin⁡θ<0\sin\theta < 0 and cos⁡θ<0\cos\theta < 0
  7. Misc II Q3(iii)
    sin⁡θ>0\sin\theta > 0 and tan⁡θ<0\tan\theta < 0
  8. Misc II Q4
    Which is greater sin⁡(1856∘)\sin(1856^\circ) or sin⁡(2006∘)\sin(2006^\circ) ?
  9. Misc II Q5
    Which of the following is positive ? sin⁡(−310∘)\sin(-310^\circ) or sin⁡(310∘)\sin(310^\circ)
  10. Misc II Q6
    Show that 1−2sin⁡θ cos⁡θ≥01- 2\sin\theta\, \cos\theta \ge 0 for all θ∈R\theta \in R
  11. Misc II Q7
    Show that tan⁡2θ+cot⁡2θ≥2\tan^2\theta + \cot^2\theta \ge 2 for all θ∈R\theta \in R
  12. Misc II Q8
    If sin⁡θ=x2−y2x2+y2\sin\theta = \frac{x^2-y^2}{x^2+y^2} then find the values of cos⁡θ\cos\theta, tan⁡θ\tan\theta in terms of xx and yy.
  13. Misc II Q9
    If sec⁡θ=2\sec\theta = \sqrt{2} and 3π2<θ<2π\frac{3\pi}{2} < \theta < 2\pi then evaluate 1+tan⁡θ+cosec⁡θ1+cot⁡θ−cosec⁡θ\frac{1+\tan\theta+\operatorname{cosec}\theta}{1+\cot\theta-\operatorname{cosec}\theta}
  14. Prove the following:
    Misc II Q10(i)
    sin⁡2A cos⁡2B+cos⁡2A sin⁡2B+cos⁡2A cos⁡2B+sin⁡2A sin⁡2B=1\sin^2 A\, \cos^2 B + \cos^2 A\, \sin^2 B + \cos^2 A\, \cos^2 B + \sin^2 A\, \sin^2 B = 1
  15. Misc II Q10(ii)
    (1+cot⁡θ+tan⁡θ)(sin⁡θ−cos⁡θ)sec⁡3θ−cosec⁡3θ=sin⁡2θ cos⁡2θ\frac{(1+\cot\theta+\tan\theta)(\sin\theta-\cos\theta)}{\sec^3\theta-\operatorname{cosec}^3\theta} = \sin^2\theta\, \cos^2\theta
  16. Misc II Q10(iii)
    (tan⁡θ+1cos⁡θ)2+(tan⁡θ−1cos⁡θ)2=2(1+sin⁡2θ1−sin⁡2θ)\left(\tan\theta+\frac{1}{\cos\theta}\right)^2 + \left(\tan\theta-\frac{1}{\cos\theta}\right)^2 = 2\left(\frac{1+\sin^2\theta}{1-\sin^2\theta}\right)
  17. Misc II Q10(iv)
    2sec⁡2θ−sec⁡4θ−2cosec⁡2θ+cosec⁡4θ=cot⁡4θ−tan⁡4θ2\sec^2\theta - \sec^4\theta - 2\operatorname{cosec}^2\theta + \operatorname{cosec}^4\theta = \cot^4\theta - \tan^4\theta
  18. Misc II Q10(v)
    sin⁡4θ+cos⁡4θ=1−2sin⁡2θ cos⁡2θ\sin^4\theta + \cos^4\theta = 1 - 2\sin^2\theta\, \cos^2\theta
  19. Misc II Q10(vi)
    2(sin⁡6θ+cos⁡6θ)−3(sin⁡4θ+cos⁡4θ)+1=02(\sin^6\theta + \cos^6\theta) - 3(\sin^4\theta + \cos^4\theta) + 1 = 0
  20. Misc II Q10(vii)
    cos⁡4θ−sin⁡4θ+1=2cos⁡2θ\cos^4\theta - \sin^4\theta + 1 = 2\cos^2\theta
  21. Misc II Q10(viii)
    sin⁡4θ+2sin⁡2θ⋅cos⁡2θ=1−cos⁡4θ\sin^4\theta + 2\sin^2\theta \cdot \cos^2\theta = 1 - \cos^4\theta
  22. Misc II Q10(ix)
    sin⁡3θ+cos⁡3θsin⁡θ+cos⁡θ+sin⁡3θ−cos⁡3θsin⁡θ−cos⁡θ=2\frac{\sin^3\theta + \cos^3\theta}{\sin\theta + \cos\theta} + \frac{\sin^3\theta - \cos^3\theta}{\sin\theta - \cos\theta} = 2
  23. Misc II Q10(x)
    tan⁡2θ−sin⁡2θ=sin⁡4θ sec⁡2θ\tan^2\theta - \sin^2\theta = \sin^4\theta\, \sec^2\theta
  24. Misc II Q10(xi)
    (sin⁡θ+cosec⁡θ)2+(cos⁡θ+sec⁡θ)2=tan⁡2θ+cot⁡2θ+7(\sin\theta + \operatorname{cosec}\theta)^2 + (\cos\theta + \sec\theta)^2 = \tan^2\theta + \cot^2\theta + 7
  25. Misc II Q10(xii)
    sin⁡8θ−cos⁡8θ=(sin⁡2θ−cos⁡2θ)(1−2sin⁡2θ cos⁡2θ)\sin^8\theta - \cos^8\theta = (\sin^2\theta - \cos^2\theta)(1 - 2\sin^2\theta\, \cos^2\theta)
  26. Misc II Q10(xiii)
    sin⁡6A+cos⁡6A=1−3sin⁡2A+3sin⁡4A\sin^6 A + \cos^6 A = 1 - 3\sin^2 A + 3\sin^4 A
  27. Misc II Q10(xiv)
    (1+tan⁡A⋅tan⁡B)2+(tan⁡A−tan⁡B)2=sec⁡2A⋅sec⁡2B(1 + \tan A \cdot \tan B)^2 + (\tan A - \tan B)^2 = \sec^2 A \cdot \sec^2 B
  28. Misc II Q10(xv)
    1+cot⁡θ+cosec⁡θ1−cot⁡θ+cosec⁡θ=cosec⁡θ+cot⁡θ−1cot⁡θ−cosec⁡θ+1\frac{1+\cot\theta+\operatorname{cosec}\theta}{1-\cot\theta+\operatorname{cosec}\theta} = \frac{\operatorname{cosec}\theta+\cot\theta-1}{\cot\theta-\operatorname{cosec}\theta+1}
  29. Misc II Q10(xvi)
    tan⁡θ+sec⁡θ−1tan⁡θ+sec⁡θ+1=tan⁡θsec⁡θ+1\frac{\tan\theta+\sec\theta-1}{\tan\theta+\sec\theta+1} = \frac{\tan\theta}{\sec\theta+1}
  30. Misc II Q10(xvii)
    cosec⁡θ+cot⁡θ−1cosec⁡θ+cot⁡θ+1=1−sin⁡θcos⁡θ\frac{\operatorname{cosec}\theta+\cot\theta-1}{\operatorname{cosec}\theta+\cot\theta+1} = \frac{1-\sin\theta}{\cos\theta}
  31. Misc II Q10(xviii)
    cosec⁡θ+cot⁡θ+1cot⁡θ+cosec⁡θ−1=cot⁡θcosec⁡θ−1\frac{\operatorname{cosec}\theta+\cot\theta+1}{\cot\theta+\operatorname{cosec}\theta-1} = \frac{\cot\theta}{\operatorname{cosec}\theta-1}