Question

if sina is a geometric mean of sinb and cosb the prove that cos2a=2cos^2 (x/4 +b)​

Answer 1

Step-by-step explanation:

If sinβ is the geometric mean between sinα and cosα, then sin

2

β=sinα.cosα

⇒cos2β=1−2sin

2

β

=1−2sinα.cosα

⇒cos

2

2β=(1−sin2α)

2

Now if we simplify option A using 2sinA.cosB=sin(A+B)+sin(A−B), we get

2sin

2

(

4

π

−α)2cos

2

(

4

π

+α)=(2sin(

4

π

−α)cos(

4

π

+α))

2

=(sin

2

π

+sin(−2α))

2

=(1−sin2α)

2

Similarly, we can verify other options.

Hence option A is correct.