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Question
Solve the following equation:
2 tan−1 (cos x) = tan−1 (2 cosec x)
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Solution
`2 tan^(-1) (cos x) = tan^(-1) (2 "cosec" x)`
⇒ `tan^(-1) ((2 cos x)/(1- cos^2 x)) = tan^(-1) (2 "cosec" x)` ...`[2 tan^(-1) x = tan^(-1) (2x)/(1 - x^2)]`
⇒ `(2 cos x)/(1 - cos^2 x) = 2 "cosec" x`
⇒`(2 cos x)/(sin^2 x) = 2/sin x`
⇒ cos x = sin x
⇒ tan x = 1
∴ `x = pi/4`
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