Q. 125.0( 2 Votes )

# Prove the following:

**OR**

Prove the following:

Answer :

Given that x ∈ (0, 1)

To Prove:

Proof: Take Right Hand Side (RHS),

Substitute x = tan^{2} θ in the above equation. We get

We know that,

Squaring on both sides,

We know that by trigonometric identity, sin^{2} θ + cos^{2} θ = 1.

Also, by trigonometric identity, cos 2θ = cos^{2} θ – sin^{2} θ.

And, cos^{-1}(cos 2θ) = 2θ

⇒ RHS = θ …(i)

We had assumed x = tan^{2} θ

If x = tan^{2} 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,

sin^{2} x + cos^{2} x = 1

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

sin^{2} x = 1 – cos^{2} x

Substituting the value of cos x from (i),

…(iii)

Also,

sin^{2} y + cos^{2} y = 1

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

cos^{2} y = 1 – sin^{2} 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.**

Rate this question :

Prove the following:

**OR**

Prove the following:

Mathematics - Board Papers

Solve the following equation for x :

Mathematics - Board Papers