Question 1187233
here's a summary of trig identities that can help.


<a href = "https://www2.clarku.edu/faculty/djoyce/trig/identities.html" target = "_blank">https://www2.clarku.edu/faculty/djoyce/trig/identities.html</a>


the trig identity you want is:


cos(2*theta) = cos^2(theta) - sin^2(theta)


the other identity that you want is:


sin^2(theta) + cos^2(theta) = 1


in this other identity, solve for sin^2(theta) to get:


sin^2(theta) = 1 - cos^2(theta).


in the first identity of cos(2*theta) = cos^2(theta) - sin^2(theta), replace sin^2(theta) with 1 - cos^2(theta) to get:


cos(2*theta) = cos^2(theta) - (1 - cos^2(theta)).


simplify to get:


cos(2*theta) = cos^2(theta - 1 + cos^2(theta)


combine like terms to get:


cos(2*theta) = 2cos^2(theta) - 1


note that 2cos^2(theta) is the same as 2cos(theta)^2.


you get:


cos(2*theta) = 2cos(theta)^2 - 1


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


trust me that cos^2(x) is equal to cos(x)^2.
if you tried to do cos^2(x) in your calculator, it wouldn't be able to do it.
you would have to enter cos^2(x) as cos(x)^2.
at least that's the way it is with my ti-84 plus calculator.