Question

Show all steps necessary to verify the trigonometric identity: 1 - 2cos2 x = 2sin2 x - 1

Answer 1

ANSWER

See explanation.

EXPLANATION

We want to very the identity:

$1 - 2 \cos^{2} (x) = 2 \sin^{2} (x) - 1$

Let us take the Left hand side and work it out to get the right hand side.

$1 - 2 \cos^{2} (x)$

Recall the Pythagorean Identity:

$\cos^{2} (x) + \sin^{2} (x) = 1$

Make cos²(x) the subject.

$\cos^{2} (x) = 1 - \sin^{2} (x)$

We substitute this into our expression to get:

$1 - 2 (1 - \sin^{2} (x) )$

Expand:

$1 - 2 + 2\sin^{2} (x)$

Simplify

$- 1+ 2\sin^{2} (x)$

We can rewrite to get;

$2\sin^{2} (x) - 1$

Therefore:

$1 - 2 \cos^{2} (x) = 2 \sin^{2} (x) - 1$