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Value of tan, cos or cosec of a sum or multiple of inverse trigonometric values

This type of question, asked 6 times: 2021 Paper 4, 2021 Paper 12, 29 Jun 2022, 26 Jul 2022, 28 Jan 2025, 28 Jan 2026 S2. All 6 questions below, oldest first.

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  1. Question 1

    2021 Paper 4

    79
    csc⁡[2cot⁡−1(5)+cos⁡−1(45)]\csc\left\lbrack 2\cot^{- 1}(5) +\cos^{- 1}\left( \frac{4}{5} \right) \right\rbrack is equal to:

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  2. Question 2

    2021 Paper 12

    40
    The value of tan⁡(2tan⁡−1(35)+sin⁡−1(513))\tan\left( 2\tan^{- 1}\left( \frac{3}{5} \right)+\sin^{- 1}\left( \frac{5}{13} \right) \right) is equal to :

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  3. Question 3

    29 Jun 2022

    86
    50tan⁡(3tan⁡−1(12)+2cos⁡−1(15))+50\tan\left( 3\tan^{- 1}\left( \frac{1}{2} \right)+ 2\cos^{- 1}\left( \frac{1}{\sqrt{5}} \right) \right)+ 42tan⁡(12tan⁡−1(22))4\sqrt{2}\tan\left( \frac{1}{2}\tan^{- 1}(2\sqrt{2}) \right) is equal to

  4. Question 4

    26 Jul 2022

    79
    tan⁡(2tan⁡−115+sec⁡−152+2tan⁡−118)\tan\left( 2\tan^{- 1}\frac{1}{5}+\sec^{- 1}\frac{\sqrt{5}}{2}+ 2\tan^{- 1}\frac{1}{8} \right) is equal to:

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  5. Question 5

    28 Jan 2025

    60
    cos⁡(sin⁡−135+sin⁡−1513+sin⁡−13365)\cos\left( \sin^{- 1}\frac{3}{5}+\sin^{- 1}\frac{5}{13}+\sin^{- 1}\frac{33}{65} \right) is equal to :

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  6. Question 6

    28 Jan 2026 S2

    56
    Considering the principal values of inverse trigonometric functions, the value of the expression tan⁡(2sin⁡−1(213)−2cos⁡−1(310))\tan\left( 2\sin^{- 1}\left( \frac{2}{\sqrt{13}} \right)- 2\cos^{- 1}\left( \frac{3}{\sqrt{10}} \right) \right) is equal to :

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