PYQs Guides Notes Mocks Board Questions NDA Mathematics Trigonometric Identities 145 NDA Mathematics previous year questions (PYQs) from 19 papers, 2017 to 2026, with answers and worked solutions.
About this chapter Difficulty: 33 easy · 65 moderate · 47 hard. Most-asked subtopics: Compound Angle Formulas (42), Multiple and Half-Angle Formulas (31), Product-to-Sum and Sum-to-Product Identities (27). Updated 14 September 2026. For the next two (02) items that follow :
The angles A A A , B B B and C C C of a triangle A B C ABC A B C are in the ratio 1 : 2 : 7 1 : 2 : 7 1 : 2 : 7 . #1 · Specific Values and Quadrants
What is sin A ⋅ cos B \sin A \cdot \cos B sin A ⋅ cos B equal to? #2 · Specific Values and Quadrants
What is
tan ( 1125 ∘ ) cot ( 405 ∘ ) + tan ( 765 ∘ ) cot ( 675 ∘ ) \tan(1125^\circ) \cot(405^\circ) + \tan(765^\circ) \cot(675^\circ) tan ( 112 5 ∘ ) cot ( 40 5 ∘ ) + tan ( 76 5 ∘ ) cot ( 67 5 ∘ )
equal to? #3 · Specific Values and Quadrants
What is the value of tan 18 ° \tan 18° tan 18° ? A 5 − 1 10 + 2 5 \dfrac{\sqrt{5}-1}{\sqrt{10+2\sqrt{5}}} 10 + 2 5 5 − 1 B 5 − 1 10 + 5 \dfrac{\sqrt{5}-1}{\sqrt{10+\sqrt{5}}} 10 + 5 5 − 1 C 10 + 2 5 5 − 1 \dfrac{\sqrt{10+2\sqrt{5}}}{\sqrt{5}-1} 5 − 1 10 + 2 5 D 10 + 5 5 − 1 \dfrac{\sqrt{10+\sqrt{5}}}{\sqrt{5}-1} 5 − 1 10 + 5 #4 · Specific Values and Quadrants
If sec θ − cosec θ = 4 3 \sec\theta - \text{cosec}\,\theta = \dfrac{4}{3} sec θ − cosec θ = 3 4 , then what is ( sin θ − cos θ ) (\sin\theta - \cos\theta) ( sin θ − cos θ ) equal to? A B C Both − 2 -2 − 2 and 1 2 \dfrac{1}{2} 2 1 D Neither 1 2 \dfrac{1}{2} 2 1 nor − 2 -2 − 2 #5 · Specific Values and Quadrants
The value of 3 cosec 20 ° − sec 20 ° \sqrt{3}\,\text{cosec}\,20° - \sec 20° 3 cosec 20° − sec 20° is equal to Done a few? Here’s your next move
#6 · Specific Values and Quadrants
1 + sin A = − ( sin A 2 + cos A 2 ) \sqrt{1+\sin A} = -\!\left(\sin\dfrac{A}{2} + \cos\dfrac{A}{2}\right) 1 + sin A = − ( sin 2 A + cos 2 A ) is true if A 3 π 2 < A < 5 π 2 \dfrac{3\pi}{2} < A < \dfrac{5\pi}{2} 2 3 π < A < 2 5 π only B π 2 < A < 3 π 2 \dfrac{\pi}{2} < A < \dfrac{3\pi}{2} 2 π < A < 2 3 π only C 3 π 2 < A < 7 π 2 \dfrac{3\pi}{2} < A < \dfrac{7\pi}{2} 2 3 π < A < 2 7 π D 0 < A < 3 π 2 0 < A < \dfrac{3\pi}{2} 0 < A < 2 3 π #7 · Specific Values and Quadrants
If sin θ = − 1 2 \sin\theta=-\dfrac{1}{2} sin θ = − 2 1 and tan θ = 1 3 \tan\theta=\dfrac{1}{\sqrt{3}} tan θ = 3 1 , then in which quadrant does θ \theta θ lie? #8 · Specific Values and Quadrants
If cosec θ = 29 21 \cosec\theta = \dfrac{29}{21} cosec θ = 21 29 where 0 < θ < 90 ° 0 < \theta < 90° 0 < θ < 90° , then what is the value of 4 sec θ + 4 tan θ 4\sec\theta + 4\tan\theta 4 sec θ + 4 tan θ ? #9 · Specific Values and Quadrants
What is cos 80 ° + cos 40 ° − cos 20 ° \cos80° + \cos40° - \cos20° cos 80° + cos 40° − cos 20° equal to ? Directions for the following three (03) items: Read the following information and answer the three items that follow: Let α = β = 15 ° \alpha = \beta = 15° α = β = 15° . #10 · Specific Values and Quadrants
What is the value of sin α + cos β \sin\alpha + \cos\beta sin α + cos β ? A B 1 2 2 \frac{1}{2\sqrt{2}} 2 2 1 C 3 2 2 \frac{\sqrt{3}}{2\sqrt{2}} 2 2 3 D 3 2 \frac{\sqrt{3}}{\sqrt{2}} 2 3 #11 · Specific Values and Quadrants
What is the value of sin 7 α − cos 7 β \sin 7\alpha - \cos 7\beta sin 7 α − cos 7 β ? A B 1 2 2 \frac{1}{2\sqrt{2}} 2 2 1 C 3 2 2 \frac{\sqrt{3}}{2\sqrt{2}} 2 2 3 D 3 2 \frac{\sqrt{3}}{\sqrt{2}} 2 3 #12 · Specific Values and Quadrants
What is sin ( α + 1 ° ) + cos ( β + 1 ° ) \sin(\alpha+1°)+\cos(\beta+1°) sin ( α + 1° ) + cos ( β + 1° ) equal to? A 3 cos 1 ° + sin 1 ° \sqrt{3}\cos 1°+\sin 1° 3 cos 1° + sin 1° B 3 cos 1 ° − 1 2 sin 1 ° \sqrt{3}\cos 1°-\frac{1}{2}\sin 1° 3 cos 1° − 2 1 sin 1° C 1 2 ( 3 cos 1 ° − sin 1 ° ) \frac{1}{\sqrt{2}}(\sqrt{3}\cos 1°-\sin 1°) 2 1 ( 3 cos 1° − sin 1° ) D 1 2 ( 3 cos 1 ∘ + sin 1 ∘ ) \dfrac{1}{\sqrt{2}}(\sqrt{3}\cos 1^\circ+\sin 1^\circ) 2 1 ( 3 cos 1 ∘ + sin 1 ∘ ) #13 · Specific Values and Quadrants
If tan x = − 3 4 \tan x = -\frac{3}{4} tan x = − 4 3 and x x x is in the second quadrant, then what is the value of sin x ⋅ cos x \sin x \cdot \cos x sin x ⋅ cos x ? A B C D − 12 25 -\frac{12}{25} − 25 12 #14 · Specific Values and Quadrants
What is the value of csc ( 7 π 6 ) sec ( 5 π 3 ) \csc\left(\frac{7\pi}{6}\right)\sec\left(\frac{5\pi}{3}\right) csc ( 6 7 π ) sec ( 3 5 π ) ? A B C D − 4 3 -\frac{4}{\sqrt{3}} − 3 4 #15 · Specific Values and Quadrants
What is value of cot 2 15 ∘ + tan 2 15 ∘ \cot^215^\circ+\tan^215^\circ cot 2 1 5 ∘ + tan 2 1 5 ∘ ? #16 · Specific Values and Quadrants
If tan θ = − 5 12 \tan\theta=-\frac{5}{12} tan θ = − 12 5 , then what can be the value of sin θ \sin\theta sin θ ? A 5 13 \frac{5}{13} 13 5 but cannot be − 5 13 -\frac{5}{13} − 13 5 B − 5 13 -\frac{5}{13} − 13 5 but cannot be 5 13 \frac{5}{13} 13 5 C 5 13 \frac{5}{13} 13 5 or − 5 13 -\frac{5}{13} − 13 5 D #17 · Specific Values and Quadrants
What is the value of csc ( − 73 π 3 ) \csc\!\left(-\frac{73\pi}{3}\right) csc ( − 3 73 π ) ? A B − 2 3 -\frac{2}{\sqrt{3}} − 3 2 C D #18 · Specific Values and Quadrants
What is cos 36 ° − cos 72 ° \cos36° - \cos72° cos 36° − cos 72° equal to? A B − 5 2 -\frac{\sqrt{5}}{2} − 2 5 C D #19 · Specific Values and Quadrants
If sec x = 25 24 \sec x = \frac{25}{24} sec x = 24 25 and x lies in the fourth quadrant, then what is the value of tan x + sin x \tan x + \sin x tan x + sin x ? A − 625 168 -\frac{625}{168} − 168 625 B − 343 600 -\frac{343}{600} − 600 343 C 625 168 \frac{625}{168} 168 625 D 343 600 \frac{343}{600} 600 343 #20 · Specific Values and Quadrants
What is the value of tan 2 165 ° + cot 2 165 ° \tan^2 165° + \cot^2 165° tan 2 165° + cot 2 165° ? #21 · Specific Values and Quadrants
If θ \theta θ lies in the fourth quadrant and 3 cot θ + 4 = 0 3\cot\theta+4=0 3 cot θ + 4 = 0 , then what is the value of sin 2 θ + cos 2 θ \sin2\theta+\cos2\theta sin 2 θ + cos 2 θ ? A − 31 25 -\dfrac{31}{25} − 25 31 B − 17 25 -\dfrac{17}{25} − 25 17 C D #22 · Specific Values and Quadrants
What is sin 9 ° − cos 9 ° \sin9°-\cos9° sin 9° − cos 9° equal to? A − 5 − 5 2 -\dfrac{\sqrt{5-\sqrt{5}}}{2} − 2 5 − 5 B − 5 − 3 2 -\dfrac{\sqrt{5-\sqrt{3}}}{2} − 2 5 − 3 C 5 − 5 2 \dfrac{\sqrt{5-\sqrt{5}}}{2} 2 5 − 5 D 5 − 5 4 \dfrac{\sqrt{5-\sqrt{5}}}{4} 4 5 − 5 #23 · Specific Values and Quadrants
Consider the following numbers:
A. tan 22.5 ° \tan22.5° tan 22.5°
B. cot 22.5 ° \cot22.5° cot 22.5°
C. tan 22.5 ° − cot 22.5 ° \tan22.5°-\cot22.5° tan 22.5° − cot 22.5°
How many of the above are irrational numbers? For the next two (02) items that follow :
Given that tan ( A 2 + B 2 ) = p \tan\left(\frac{A}{2} + \frac{B}{2}\right) = p tan ( 2 A + 2 B ) = p and tan ( A 2 − B 2 ) = q \tan\left(\frac{A}{2} - \frac{B}{2}\right) = q tan ( 2 A − 2 B ) = q , where p q ≠ ± 1 pq \ne \pm 1 pq = ± 1 . #24 · Compound Angle Formulas
What is tan A \tan A tan A equal to? A p − q 1 + p q \frac{p - q}{1 + pq} 1 + pq p − q B p + q 1 − p q \frac{p + q}{1 - pq} 1 − pq p + q C p + q 1 + p q \frac{p + q}{1 + pq} 1 + pq p + q D p − q 1 − p q \frac{p - q}{1 - pq} 1 − pq p − q #25 · Compound Angle Formulas
What is tan B \tan B tan B equal to? A p − q 1 + p q \frac{p - q}{1 + pq} 1 + pq p − q B p + q 1 − p q \frac{p + q}{1 - pq} 1 − pq p + q C p + q 1 + p q \frac{p + q}{1 + pq} 1 + pq p + q D p − q 1 − p q \frac{p - q}{1 - pq} 1 − pq p − q For the next two (02) items that follow :
Let 4 ( A + B ) = π 4(A + B) = \pi 4 ( A + B ) = π . #26 · Compound Angle Formulas
What is ( 1 + tan A ) ( 1 + tan B ) (1 + \tan A)(1 + \tan B) ( 1 + tan A ) ( 1 + tan B ) equal to? #27 · Compound Angle Formulas
What is ( cot A − 1 ) ( cot B − 1 ) (\cot A - 1)(\cot B - 1) ( cot A − 1 ) ( cot B − 1 ) equal to? #28 · Compound Angle Formulas
If cot α \cot\alpha cot α and cot β \cot\beta cot β are the roots of the equation x 2 + b x + c = 0 x^2 + bx + c = 0 x 2 + b x + c = 0 with b ≠ 0 b \neq 0 b = 0 , then the value of cot ( α + β ) \cot(\alpha+\beta) cot ( α + β ) is #29 · Compound Angle Formulas
If sin θ = 3 sin ( θ + 2 α ) \sin\theta = 3\sin(\theta+2\alpha) sin θ = 3 sin ( θ + 2 α ) , then the value of tan ( θ + α ) + 2 tan α \tan(\theta+\alpha)+2\tan\alpha tan ( θ + α ) + 2 tan α is equal to #30 · Compound Angle Formulas
The value of tan 9 ° − tan 27 ° − tan 63 ° + tan 81 ° \tan 9° - \tan 27° - \tan 63° + \tan 81° tan 9° − tan 27° − tan 63° + tan 81° is equal to #31 · Compound Angle Formulas
What is sin 105 ° + cos 105 ° \sin 105° + \cos 105° sin 105° + cos 105° equal to? #32 · Compound Angle Formulas
If sin ( x + y ) sin ( x − y ) = a + b a − b \frac{\sin(x+y)}{\sin(x-y)} = \frac{a+b}{a-b} s i n ( x − y ) s i n ( x + y ) = a − b a + b , then what is tan x tan y \frac{\tan x}{\tan y} t a n y t a n x equal to? A B C a + b a − b \frac{a+b}{a-b} a − b a + b D a − b a + b \frac{a-b}{a+b} a + b a − b #33 · Compound Angle Formulas
If cos α + cos β + cos γ = 0 \cos\alpha + \cos\beta + \cos\gamma = 0 cos α + cos β + cos γ = 0 , where 0 < α , β , γ ≤ π / 2 0 < \alpha, \beta, \gamma \le \pi/2 0 < α , β , γ ≤ π /2 , then what is sin α + sin β + sin γ \sin\alpha + \sin\beta + \sin\gamma sin α + sin β + sin γ ? A B C 5 2 2 \frac{5\sqrt{2}}{2} 2 5 2 D 3 2 2 \frac{3\sqrt{2}}{2} 2 3 2 #34 · Compound Angle Formulas
If sec ( θ − α ) \sec(\theta-\alpha) sec ( θ − α ) , sec θ \sec\theta sec θ and sec ( θ + α ) \sec(\theta+\alpha) sec ( θ + α ) are in AP, where cos α ≠ 1 \cos\alpha \neq 1 cos α = 1 , then what is the value of sin 2 θ + cos α \sin^2\theta + \cos\alpha sin 2 θ + cos α ? #35 · Compound Angle Formulas
What is the value of sin 34 ° cos 236 ° − sin 56 ° sin 124 ° cos 28 ° cos 88 ° + cos 178 ° sin 208 ° \dfrac{\sin 34° \cos 236° - \sin 56° \sin 124°}{\cos 28° \cos 88° + \cos 178° \sin 208°} cos 28° cos 88° + cos 178° sin 208° sin 34° cos 236° − sin 56° sin 124° ? #36 · Compound Angle Formulas
tan 54 ° \tan 54° tan 54° can be expressed as A sin 9 ° + cos 9 ° sin 9 ° − cos 9 ° \dfrac{\sin 9° + \cos 9°}{\sin 9° - \cos 9°} sin 9° − cos 9° sin 9° + cos 9° B sin 9 ° − cos 9 ° sin 9 ° + cos 9 ° \dfrac{\sin 9° - \cos 9°}{\sin 9° + \cos 9°} sin 9° + cos 9° sin 9° − cos 9° C cos 9 ° + sin 9 ° cos 9 ° − sin 9 ° \dfrac{\cos 9° + \sin 9°}{\cos 9° - \sin 9°} cos 9° − sin 9° cos 9° + sin 9° D sin 36 ° cos 36 ° \dfrac{\sin 36°}{\cos 36°} cos 36° sin 36° Consider the following for the next 02 (two) items:
It is given that cos ( θ − α ) = a \cos(\theta-\alpha)=a cos ( θ − α ) = a , cos ( θ − β ) = b \cos(\theta-\beta)=b cos ( θ − β ) = b . #37 · Compound Angle Formulas
What is cos ( α − β ) \cos(\alpha-\beta) cos ( α − β ) equal to? A a b + 1 − a 2 1 − b 2 ab + \sqrt{1-a^2}\sqrt{1-b^2} ab + 1 − a 2 1 − b 2 B a b − 1 − a 2 1 − b 2 ab - \sqrt{1-a^2}\sqrt{1-b^2} ab − 1 − a 2 1 − b 2 C a 1 − b 2 − b 1 − a 2 a\sqrt{1-b^2} - b\sqrt{1-a^2} a 1 − b 2 − b 1 − a 2 D a 1 − b 2 + b 1 − a 2 a\sqrt{1-b^2} + b\sqrt{1-a^2} a 1 − b 2 + b 1 − a 2 #38 · Compound Angle Formulas
What is sin 2 ( α − β ) + 2 a b cos ( α − β ) \sin^2(\alpha-\beta)+2ab\cos(\alpha-\beta) sin 2 ( α − β ) + 2 ab cos ( α − β ) equal to? A B C D − ( a 2 + b 2 ) -(a^2+b^2) − ( a 2 + b 2 ) #39 · Compound Angle Formulas
What is the value of tan 75 ° + cot 75 ° \tan 75° + \cot 75° tan 75° + cot 75° ? #40 · Compound Angle Formulas
What is ( 1 + tan α tan β ) 2 + ( tan α − tan β ) 2 − sec 2 α sec 2 β (1+\tan\alpha\tan\beta)^2+(\tan\alpha-\tan\beta)^2-\sec^2\alpha\sec^2\beta ( 1 + tan α tan β ) 2 + ( tan α − tan β ) 2 − sec 2 α sec 2 β equal to? #41 · Compound Angle Formulas
If sin θ + cos θ = 2 cos θ \sin\theta+\cos\theta=\sqrt{2}\cos\theta sin θ + cos θ = 2 cos θ , then what is ( cos θ − sin θ ) (\cos\theta-\sin\theta) ( cos θ − sin θ ) equal to? A − 2 cos θ -\sqrt{2}\cos\theta − 2 cos θ B − 2 sin θ -\sqrt{2}\sin\theta − 2 sin θ C 2 sin θ \sqrt{2}\sin\theta 2 sin θ D #42 · Compound Angle Formulas
If tan A − tan B = x \tan A - \tan B = x tan A − tan B = x and cot B − cot A = y \cot B - \cot A = y cot B − cot A = y , then what is the value of cot ( A − B ) \cot(A - B) cot ( A − B ) ? A 1 x + 1 y \dfrac{1}{x} + \dfrac{1}{y} x 1 + y 1 B 1 y − 1 x \dfrac{1}{y} - \dfrac{1}{x} y 1 − x 1 C x y x + y \dfrac{xy}{x+y} x + y x y D 1 + 1 x y 1 + \dfrac{1}{xy} 1 + x y 1 #43 · Compound Angle Formulas
If tan θ = cos 17 ° − sin 17 ° cos 17 ° + sin 17 ° \tan\theta=\frac{\cos 17°-\sin 17°}{\cos 17°+\sin 17°} tan θ = c o s 17° + s i n 17° c o s 17° − s i n 17° , then what is the value of θ \theta θ ? Directions for the following three (03) items: Read the following information and answer the three items that follow: Let a sin 2 x + b cos 2 x = c a\sin^2 x + b\cos^2 x = c a sin 2 x + b cos 2 x = c ; b sin 2 y + a cos 2 y = d b\sin^2 y + a\cos^2 y = d b sin 2 y + a cos 2 y = d and p tan x = q tan y p\tan x = q\tan y p tan x = q tan y . #44 · Compound Angle Formulas
What is tan 2 x \tan^2 x tan 2 x equal to? A c − b a − c \frac{c-b}{a-c} a − c c − b B a − c c − b \frac{a-c}{c-b} c − b a − c C c − a c − b \frac{c-a}{c-b} c − b c − a D c − b c − a \frac{c-b}{c-a} c − a c − b #45 · Compound Angle Formulas
What is d − a b − d \frac{d-a}{b-d} b − d d − a equal to? #46 · Compound Angle Formulas
What is p 2 q 2 \frac{p^2}{q^2} q 2 p 2 equal to? A ( b − c ) ( b − d ) ( a − d ) ( a − c ) \frac{(b-c)(b-d)}{(a-d)(a-c)} ( a − d ) ( a − c ) ( b − c ) ( b − d ) B ( a − d ) ( c − a ) ( b − c ) ( d − b ) \frac{(a-d)(c-a)}{(b-c)(d-b)} ( b − c ) ( d − b ) ( a − d ) ( c − a ) C ( d − a ) ( c − a ) ( b − c ) ( d − b ) \frac{(d-a)(c-a)}{(b-c)(d-b)} ( b − c ) ( d − b ) ( d − a ) ( c − a ) D ( b − c ) ( b − d ) ( c − a ) ( a − d ) \frac{(b-c)(b-d)}{(c-a)(a-d)} ( c − a ) ( a − d ) ( b − c ) ( b − d ) #47 · Compound Angle Formulas
What is the value of tan 31 ° tan 33 ° tan 35 ° ⋯ tan 57 ° tan 59 ° \tan 31° \tan 33° \tan 35° \cdots \tan 57° \tan 59° tan 31° tan 33° tan 35° ⋯ tan 57° tan 59° ? #48 · Compound Angle Formulas
What is the value of ( sin 24 ° + cos 66 ° ) ( sin 24 ° − cos 66 ° ) (\sin 24° + \cos 66°)(\sin 24° - \cos 66°) ( sin 24° + cos 66° ) ( sin 24° − cos 66° ) ? #49 · Compound Angle Formulas
What is ( 1 + cot θ − csc θ ) ( 1 + tan θ + sec θ ) (1+\cot\theta - \csc\theta)(1+\tan\theta+\sec\theta) ( 1 + cot θ − csc θ ) ( 1 + tan θ + sec θ ) equal to? #50 · Compound Angle Formulas
What is 1 + tan 2 θ 1 + cot 2 θ − ( 1 − tan θ 1 − cot θ ) 2 \frac{1+\tan^2\theta}{1+\cot^2\theta} - \left(\frac{1-\tan\theta}{1-\cot\theta}\right)^2 1 + c o t 2 θ 1 + t a n 2 θ − ( 1 − c o t θ 1 − t a n θ ) 2 equal to? Showing the first 50 of 145.
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