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Turn nested expressions like sin(tan^-1 x) or cos(cot^-1 x) into algebra with a right triangle

This type of question, asked 4 times: 2021 Paper 18, 27 Jul 2022, 24 Jan 2025, 6 Apr 2026 S2. All 4 questions below, oldest first.

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  1. Question 1

    2021 Paper 18

    67
    Let f(x)=cos⁡(2tan⁡−1sin⁡(cot⁡−11−xx))f(x) = \cos\left( 2\tan^{- 1}\sin\left( \cot^{- 1}\sqrt{\frac{1 -x}{x}} \right) \right) 0<x<10 < x < 1. Then:

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  2. Question 2

    27 Jul 2022

    81
    For k∈Rk \in R, let the solutions of the equation
    cos⁡(sin⁡−1(xcot⁡(tan⁡−1(cos⁡(sin⁡−1x)))))=k,0<∣x∣<12\cos\left( \sin^{- 1}\left( x\cot\left( \tan^{- 1}\left( \cos\left( \sin^{- 1}x \right) \right) \right) \right) \right)= k,0 < |x| <\frac{1}{\sqrt{2}}
    be α\alpha and β\beta, where the inverse trigonometric functions take only principal values. If the solutions of the equation x2−bx−5=0x^{2}- bx - 5 = 0 are 1α2+1β2\frac{1}{\alpha^{2}}+\frac{1}{\beta^{2}} and αβ\frac{\alpha}{\beta}, then bk2\frac{b}{k^{2}} is equal to

  3. Question 3

    24 Jan 2025

    72
    If for some α,β;α≤β,α+β=8\alpha,\beta;\alpha \leq \beta,\alpha + \beta = 8 and sec⁡2(tan⁡−1α)+csc⁡2(cot⁡−1β)=36\sec^{2}\left( \tan^{- 1}\alpha \right)+\csc^{2}\left( \cot^{- 1}\beta \right)= 36, then α2+β\alpha^{2}+ \beta is  ____\ \_\_\_\_ .

  4. Question 4

    6 Apr 2026 S2

    63
    If sin⁡(tan⁡−1(x2))=cot⁡(sin⁡−11−x2),x∈(0,1)\sin\left( \tan^{- 1}(x\sqrt{2}) \right)= \cot\left( \sin^{- 1}\sqrt{1 -x^{2}} \right),x \in (0,1), then the value of x is :

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