# Prove the following:ORProve the following:

Given that x (0, 1)

To Prove:

Proof: Take Right Hand Side (RHS),

Substitute x = tan2 θ in the above equation. We get

We know that,

Squaring on both sides,

We know that by trigonometric identity, sin2 θ + cos2 θ = 1.

Also, by trigonometric identity, cos 2θ = cos2 θ – sin2 θ.

And, cos-1(cos 2θ) = 2θ

RHS = θ …(i)

We had assumed x = tan2 θ

If x = tan2 θ

x = (tan θ)2

√x = tan θ

θ = tan-1 √x

So, substituting this value of θ in equation (i), we get

RHS = tan-1 √x

RHS = LHS

Hence, proved.

OR

We are given with a trigonometric equation.

To Prove:

Proof: Take Left Hand Side (LHS),

Let

…(i)

And let

…(ii)

Let us also find sin x and cos y.

We know the trigonometric identity,

sin2 x + cos2 x = 1

We need to find the value of sin x. So,

sin2 x = 1 – cos2 x

Substituting the value of cos x from (i),

…(iii)

Also,

sin2 y + cos2 y = 1

We need to find the value of cos y. So,

cos2 y = 1 – sin2 y

Substituting the value of sin y from (ii),

…(iv)

Now, we have the values of sin x, cos x, sin y and cos y.

Using trigonometric identity,

sin (x + y) = sin x cos y + cos x sin y

Substituting values of sin x, cos x, sin y and cos y from eq. (i), (ii), (iii) and (iv) respectively.

Substituting values of x and y,

LHS = RHS

Hence, proved.

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