Question

Rewrite with only sin x and cos x.

cos 2x + sin x

1 + 3 sin x
1 + 3 sin2x
1 - 2 sin2x + sin x
1 + 2 sin2x + sin x

Answer 1

`cos(2x)+sin(x)=1-2\,sin^2(x)+sin(x)`

which agrees with the third option listed.

Explanation:

I believe you meant to write the answer with only sin(x).

Use the double angle identity for the cos(2x) as shown below:

`cos(2x) = cos^2(x)- sin^2(x)\\since \\cos^2(x) = 1-sin^2(x)\\then:\\cos(2x) = 1-sin^2(x)- sin^2(x)\\cos(2x)=1 - 2\,sin^2(x)`

Now we use this identity in the original expression to get:

`cos(2x)+sin(x)=1-2\,sin^2(x)+sin(x)`

which agrees with the third option listed.