Question

If cosx = 1/4, then, [sin(x/2) + cos2x] is equal to – 

1. (2√5 + 5)/8
2. (√5 + 7)/8
3. (2√5 + 7)/8
4. (2√5 - 7)/8

Answer 1

Correct Answer - Option 3 : (2√5 + 7)/8

Given:

cosx = ¼

Formula used:

cos2x = 2sin²x – 1 = 1 – 2cos²x

Also, cosx = 2sin²(x/2) - 1

Calculation:

cosx = 2sin²(x/2) - 1

⇒ ¼ = 2sin²(x/2) - 1

⇒ 2sin²(x/2) = (1/4 + 1)

⇒ sin²(x/2) = 5/4

⇒ sin(x/2) = √(5/4)

cos2x = 1 – 2cos²x = 1 – 2(1/4)²

⇒ cos2x = 7/8

According to question –

[sin(x/2) + cos2x]

⇒ √5/4 + 7/8

⇒ (2√5 + 7)/8