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Eliminate the angle: square and add or subtract using a Pythagorean identity

This type of question, asked 8 times: 17 Jul 2023 S3, 19 Jul 2023 S2, 27 Jul 2023 S2, 13 Sep 2024 S3, 17 Sep 2024 S2, 23 Sep 2024 S1, 25 Sep 2024 S2, 26 Sep 2024 S3. All 8 questions below, oldest first.

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  1. Question 1

    17 Jul 2023 S3

    51
    If acot⁡θ+b cosec θ=pa \cot \theta + b\,\text{cosec}\, \theta = p and bcot⁡θ+a cosec θ=qb \cot \theta + a\,\text{cosec}\, \theta = q then p2−q2p^2 - q^2 is equal to _____.

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  2. Question 2

    19 Jul 2023 S2

    62
    If P cos α\alpha = 3 and 4 tan α\alpha = Q, then what is the relation between P and Q, which is independent of α\alpha?

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  3. Question 3

    27 Jul 2023 S2

    60
    If 3x=sec⁡A3x = \sec A and 3x=tan⁡A\frac{3}{x} = \tan A then 9(x2−1x2)9\left(x^2 - \frac{1}{x^2}\right) is _______.

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  4. Question 4

    13 Sep 2024 S3

    70
    If m=asec⁡Am = a\sec A and y=btan⁡Ay = b\tan A, then find the value of b2m2−a2y2+a2y2b2m2+cos⁡2Ab^2m^2 - a^2y^2 + \frac{a^2y^2}{b^2m^2} + \cos^2 A.

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  5. Question 5

    17 Sep 2024 S2

    56
    If a=xcos⁡θ+ysin⁡θa = x\cos\theta + y\sin\theta and b=xsin⁡θ−ycos⁡θb = x\sin\theta - y\cos\theta, then a2+b2a^2 + b^2 is equal to:

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  6. Question 6

    23 Sep 2024 S1

    74
    Let x=rcos⁡(t)x = r \cos (t), y=rsin⁡(t)cos⁡(u)y = r \sin (t) \cos (u), z=rsin⁡(t)sin⁡(u)z = r \sin (t)\sin (u). Then the value of x2+y2+z2x^2 + y^2 + z^2 is ___.

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  7. Question 7

    25 Sep 2024 S2

    72
    If x=psec⁡Acos⁡Bx = p \sec A \cos B, y=qsec⁡Asin⁡By = q \sec A \sin B and z=rtan⁡Az = r \tan A, what is the value of the following expression? x2p2+y2q2−z2r2\frac{x^2}{p^2} + \frac{y^2}{q^2} - \frac{z^2}{r^2}

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  8. Question 8

    26 Sep 2024 S3

    63
    If x=asec⁡θcos⁡ϕ,y=bsec⁡θsin⁡ϕx = a\sec\theta\cos\phi, y = b\sec\theta\sin\phi and z=ctan⁡θz = c\tan\theta, then the value of (x2a2+y2b2−z2c2)\left(\frac{x^2}{a^2} + \frac{y^2}{b^2} - \frac{z^2}{c^2}\right) is equal to:

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