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