Prove the following:

OR

Prove 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.