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Question
If 2 tan–1(cos x) = tan–1(2 cosec x). then find the value of x.
Solution
As we know that
2 tan–1(x) = `tan^-1 ((2x)/(1 - x^2))`
∴ 2 tan–1(cos x) = `tan^-1 ((2 cos x)/(1 - cos^2x))`
⇒ 2 tan–1(cos x) = `tan^-1 ((2 cos x)/(sin^2x))`
Given, 2 tan–1 (cos x) = tan–1 (2 cosec x)
∴ `tan^-1 ((2 cos x)/(sin^2x))` = tan–1 (2 cosec x) ......[From (i)]
⇒ `(2 cos x)/sin^2x `= 2 cosec x
⇒ `cosx/sin^2x = 1/sinx`
⇒ cos x = sin x
⇒ x = `π/4`
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