Question

Cos2x=2sin² x

Answer 1

cos2x=2sin² x, cos2x=1-cos2x,  2cos2x=1,  cos2x=1/2,  2x=+- П/3+2Пk, 

x=+- П/6+Пk

Answer 2

To prove the trigonometric identity cos(2x) = 2sin^2(x), we'll start with the double angle formula for cosine and the identity for sine squared.

The double angle formula for cosine states that:

cos(2x) = cos^2(x) - sin^2(x)

The identity for sine squared states that:

sin^2(x) = 1 - cos^2(x)

Substituting the identity for sine squared into the double angle formula for cosine, we have:

cos(2x) = cos^2(x) - (1 - cos^2(x))

Expanding the brackets:

cos(2x) = cos^2(x) - 1 + cos^2(x)

Combining like terms:

cos(2x) = 2cos^2(x) - 1

Now, we need to show that 2sin^2(x) is equal to 2cos^2(x) - 1:

2sin^2(x) = 2(1 - cos^2(x)) - 1 = 2 - 2cos^2(x) - 1 = 2cos^2(x) - 1

Thus, we have shown that cos(2x) = 2sin^2(x), and the trigonometric identity is proven.