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Question
If 2 tan-1(cos x) = tan-1(2 cosec x), then find the value of 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") ...[because 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
∴ x = `pi/4` ....`[because sin pi/4 = cos pi/4]`
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