Answer:
[tex]cos(2x)+sin(x)=1-2\,sin^2(x)+sin(x)[/tex]
which agrees with the third option listed.
Step-by-step 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:
[tex]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)[/tex]
Now we use this identity in the original expression to get:
[tex]cos(2x)+sin(x)=1-2\,sin^2(x)+sin(x)[/tex]
which agrees with the third option listed.
