Solve 2 tan^{- 1} (cos x) = tan^{- 1} (2cosec x)

Solution:

Inverse trigonometric functions are the inverse ratio of the basic trigonometric ratios

Here the basic trigonometric function of Sin θ = y can be changed to θ = sin^-1 y

It is given that 2 tan^{- 1} (cos x)

= tan^{- 1} (2 cosec x).

Since,

2 tan^-1 (x) = tan^-1 2x/(1 + x²)

Hence,

⇒ tan^-1 [(2 cos x)/(1- cos² x)]

= tan^-1 (2 cosec x)

⇒ (2 cos x) / (sin² x)

= 2/sin x

⇒ cos x = sin x

⇒ tan x = 1

⇒ tan x

= tan π/4

Therefore,

x = nπ + π/4, where n ∈ Z

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NCERT Solutions for Class 12 Maths - Chapter 2 Exercise ME Question 13

Solve 2 tan^{- 1} (cos x) = tan^{- 1} (2cosec x)

Summary:

Hence we have proved by using inverse trigonometric functions that 2 tan^{- 1} (cos x) = tan^{- 1} (2cosec x)