Mark the Correct alternative in the following:
If cos 2x + 2 cos x = 1 then, (2 – cos2 x) sin2 x is equal to
Given cos 2x + 2 cos x = 1, we need to find the expression,
(2 – cos2 x) sin2 x
Consider cos 2x + 2 cos x = 1
2cos2 x – 1 + 2 cos x -1 = 0
2cos2 x + 2cos x – 2 = 0
cos2 x + cos x = 1 -------- (1)
Now consider the expression
(2 – cos2 x) sin2 x = (2 – cos2 x)(1-cos2x)
= {2 – (1 – cos x)} { 1- ( 1 – cos x)}
[from equation (1) cos2 x = 1 - cos x]
= (1+ cos x) ( cos x)
= cos x + cos2 x
[from equation (1) cos2 x + cos x = 1]
= 1
Hence (2 – cos2 x) sin2 x = 1, so option A is the answer.