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Trig ratio of a sum or difference of two inverse-trig values

This type of question, asked 12 times: 2 May 2023 S1, 4 May 2023 S1, 10 May 2023 S1, 10 May 2023 S2, 11 May 2023 S1, 11 May 2023 S2, 16 May 2023 S2, 13 May 2024 S2, 22 Apr 2025 S2, 26 Apr 2025 S2, 16 Apr 2026 S1, 19 May 2026 S2. All 13 questions below, oldest first.

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  1. Question 1

    2 May 2023 S1

    105
    The value of cos⁡(2cos⁡−1x+sin⁡−1x)\cos\left(2\cos^{-1}x + \sin^{-1}x\right) at x=15x = \frac{1}{5}, where 0≤cos⁡−1x≤π0 \le \cos^{-1}x \le \pi and −π2≤sin⁡−1x≤π2-\frac{\pi}{2} \le \sin^{-1}x \le \frac{\pi}{2}, is

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  2. Question 2

    4 May 2023 S1

    107
    The value of cot⁡ ⁣(csc⁡−1 ⁣53+tan⁡−1 ⁣23)\cot\!\left(\csc^{-1}\!\frac{5}{3}+\tan^{-1}\!\frac{2}{3}\right) is

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  3. Question 3

    10 May 2023 S1

    112
    The numerical value of tan⁡ ⁣(2tan⁡−1 ⁣15+π4)\tan\!\left(2\tan^{-1}\!\frac{1}{5}+\frac{\pi}{4}\right)

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  4. Question 4

    10 May 2023 S2

    115
    tan⁡(cos⁡−112+tan⁡−112)=\tan\left(\cos^{-1}\frac{1}{\sqrt{2}}+\tan^{-1}\frac{1}{2}\right)=

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  5. Question 5

    10 May 2023 S2

    130
    The value of cos⁡(2cos⁡−1x+sin⁡−1x)\cos(2\cos^{-1}x + \sin^{-1}x) at x=15x = \frac{1}{5} is

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  6. Question 6

    11 May 2023 S1

    147
    cos⁡(sin⁡−135+cos⁡−11213)=\cos\left(\sin^{-1}\frac{3}{5} + \cos^{-1}\frac{12}{13}\right) =

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  7. Question 7

    11 May 2023 S2

    119
    The value of sin⁡(cos⁡−1(−13)−sin⁡−113)\sin\left(\cos^{-1}\left(-\frac{1}{\sqrt{3}}\right)-\sin^{-1}\frac{1}{\sqrt{3}}\right) is

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  8. Question 8

    16 May 2023 S2

    107
    If cos⁡−11213+sin⁡−135=sin⁡−1P\cos^{-1}\frac{12}{13} + \sin^{-1}\frac{3}{5} = \sin^{-1} P, then the value of PP is

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  9. Question 9

    13 May 2024 S2

    101
    The value of tan⁡ ⁣(sin⁡−1 ⁣35+tan⁡−1 ⁣23)\tan\!\left(\sin^{-1}\!\frac{3}{5}+\tan^{-1}\!\frac{2}{3}\right) is

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  10. Question 10

    22 Apr 2025 S2

    101
    Considering only the principal values of the inverse trigonometric function, the value of tan⁡(cos⁡−1152−sin⁡−1417)\tan\left( \cos^{- 1}\frac{1}{5\sqrt{2}}-\sin^{- 1}\frac{4}{\sqrt{17}} \right) is

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  11. Question 11

    26 Apr 2025 S2

    115
    Considering only the principal values of the inverse trigonometric functions, the value of tan⁡(sin⁡−1(35)−2cos⁡−1(25))\tan\left( \sin^{- 1}\left( \frac{3}{5} \right)- 2\cos^{- 1}\left( \frac{2}{\sqrt{5}} \right) \right) is

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  12. Question 12

    16 Apr 2026 S1

    67
    cos⁡(cos⁡−1(−12)+π3)=\cos\left(\cos^{-1}\left(-\dfrac{1}{2}\right) + \dfrac{\pi}{3}\right) =

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  13. Question 13

    19 May 2026 S2

    64
    The value of cot⁡(csc⁡−153+tan⁡−123)\cot\left(\mathrm{\csc}^{-1}\dfrac{5}{3} + \tan^{-1}\dfrac{2}{3}\right) is ...

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