Question

Rewrite with only sin x and cos x.

sin 2x - cos 2x

A. 2 sinx cosx - 1 + 2 sin^2x

B. 2 sin x cos^2x - 1 + 2 sin^2x

C. 2 sin x cos^2x - sin x + 1 - 2 sin^2x

D. 2 sin x cos^2x - 1 - 2 sin^2x

Answer 1

A. 2 sinx cosx - 1 + 2 sin^2 x.

Explanation:

sin2x = 2 sinx cosx

cos2x = cos^2x - sin^2x

So sin2x - cos2x = 2 sinx cosx - ( cos^2x - sin^2x)

But cos^2 x = 1 - sin^2 x, so we have:

2 sinx cosx - (1 - sin^2 x - sin^2x)

= 2 sinx cosx - 1 + 2 sin^2 x.

Answer 2

A. 2·sin(x)·cos(x) - 1 + 2·sin^2(x)

Explanation:

The double angle identities can be use directly:

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

The difference of these is ...

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