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Add several inverse-trig values of numbers into one angle (tan^-1 addition formula)

This type of question, asked 14 times: 4 May 2023 S2, 9 May 2023 S2, 15 May 2023 S1, 11 May 2024 S1, 11 May 2024 S2, 14 May 2024 S2, 21 Apr 2025 S1, 11 Apr 2026 S1, 16 Apr 2026 S1, 17 Apr 2026 S1, 19 Apr 2026 S2, 12 May 2026 S2, 14 May 2026 S1, 19 May 2026 S2. All 14 questions below, oldest first.

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  1. Question 1

    4 May 2023 S2

    119
    If cot⁡−1(7)+cot⁡−1(8)+cot⁡−1(18)=cot⁡−1x\cot^{-1}(7)+\cot^{-1}(8)+\cot^{-1}(18)=\cot^{-1}x, then the value of x is

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  2. Question 2

    9 May 2023 S2

    112
    2π−(sin⁡−145+sin⁡−1513+sin⁡−11665)2\pi - \left(\sin^{-1}\frac{4}{5} + \sin^{-1}\frac{5}{13} + \sin^{-1}\frac{16}{65}\right) is equal to

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  3. Question 3

    15 May 2023 S1

    134
    2π−(sin⁡−145+sin⁡−1513+sin⁡−11665)2\pi - \left(\sin^{-1}\frac{4}{5}+\sin^{-1}\frac{5}{13}+\sin^{-1}\frac{16}{65}\right) is equal to

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  4. Question 4

    11 May 2024 S1

    137
    The value of 2tan⁡−1 ⁣12+tan⁡−1 ⁣172\tan^{-1}\!\frac{1}{2}+\tan^{-1}\!\frac{1}{7}

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  5. Question 5

    11 May 2024 S2

    134
    π+sin⁡−1 ⁣45+sin⁡−1 ⁣513+sin⁡−1 ⁣1665\pi+\sin^{-1}\!\frac{4}{5}+\sin^{-1}\!\frac{5}{13}+\sin^{-1}\!\frac{16}{65} is equal to

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  6. Question 6

    14 May 2024 S2

    102
    The value of tan⁡−118+tan⁡−112+tan⁡−115\tan^{-1}\frac{1}{8} + \tan^{-1}\frac{1}{2} + \tan^{-1}\frac{1}{5} is

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  7. Question 7

    21 Apr 2025 S1

    123
    The value of 2tan⁡−112+tan⁡−1382\tan^{- 1}\frac{1}{2}+\tan^{- 1}\frac{3}{8} is

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  8. Question 8

    11 Apr 2026 S1

    66
    If tan⁡−1(1)+tan⁡−1(3)+tan⁡−1(5)+tan⁡−1(14)=π+tan⁡−1(α2)\tan^{-1}(1) + \tan^{-1}(3) + \tan^{-1}(5) + \tan^{-1}\left(\dfrac{1}{4}\right) = \pi + \tan^{-1}\left(\dfrac{\alpha}{2}\right), then the value of α\alpha is...

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  9. Question 9

    16 Apr 2026 S1

    68
    sin⁡−1(1213)+cos⁡−1(45)+tan⁡−1(6316)=\sin^{-1}\left(\dfrac{12}{13}\right) + \cos^{-1}\left(\dfrac{4}{5}\right) + \tan^{-1}\left(\dfrac{63}{16}\right) =

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  10. Question 10

    17 Apr 2026 S1

    67
    The value of 3tan⁡−1(12)=…3\tan^{-1}\left(\dfrac{1}{2}\right) = \ldots

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  11. Question 11

    19 Apr 2026 S2

    100
    The lengths of the shadows of a tree of height p, thrown by sun's rays at three different moments are p, 2p and 3p. The sum of the angles of elevation of the sun's rays at these three moments is equal to...

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  12. Question 12

    12 May 2026 S2

    66
    If cot⁡−12\cot^{-1}2 and cot⁡−13\cot^{-1}3 are the angles of a triangle, then the third angle is ...

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  13. Question 13

    14 May 2026 S1

    66
    2sin⁡−1(35)+tan⁡−1(15)+tan⁡−1(17)=2\sin^{-1}\left(\dfrac{3}{5}\right) + \tan^{-1}\left(\dfrac{1}{5}\right) + \tan^{-1}\left(\dfrac{1}{7}\right) =

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  14. Question 14

    19 May 2026 S2

    63
    If tan⁡−1(13)+tan⁡−1(15)+tan⁡−1(17)=π4−tan⁡−1(1k)\tan^{-1}\left(\dfrac{1}{3}\right) + \tan^{-1}\left(\dfrac{1}{5}\right) + \tan^{-1}\left(\dfrac{1}{7}\right) = \dfrac{\pi}{4} - \tan^{-1}\left(\dfrac{1}{k}\right) then the value of kk is

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