Choose the correct answer

The value of is

We need to find the value of

Let

Now, take sine on both sides,

Using the property of inverse trigonometry,

sin(sin^{-1} A) = A

Let us find the value of cos x.

We know by trigonometric identity, that

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

⇒ cos^{2} x = 1 – sin^{2} x

Put the value of sin x,

We have,

…(i)

Using the trigonometric identity,

cos 2x = cos^{2} x – sin^{2} x

⇒ cos 2x = (1 – sin^{2}x) – sin^{2} x [∵, sin^{2} x + cos^{2} x = 1]

⇒ cos 2x = 1 – sin^{2} x – sin^{2} x

⇒ cos 2x = 1 – 2 sin^{2} x

Or,

2 sin^{2} x = 1 – cos 2x

Replacing x by x/4,

Substituting the value of in equation (i),

…(ii)

Using the trigonometric identity,

cos 2x = cos^{2} x – sin^{2} x

⇒ cos 2x = cos^{2} x – (1 – cos^{2} x) [∵, sin^{2} x + cos^{2} x = 1]

⇒ cos 2x = cos^{2} x – 1 + cos^{2}x

⇒ cos 2x = 2 cos^{2} x – 1

Or,

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

Replacing x by x/2,

Substituting the value of in equation (ii),

Put the value of cos x as found above, cos x = 1/8.

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