Question

Solve the equation., and provide x values where 0 < x < 3pi: cos^2 x + cos(2x) = 5/4

Answer 1

\[\cos(2x)=\cos^2(x)-\sin^2(x)=\cos^2(x)-(1-\cos^2(x))=2\cos^2(x)-1\]
Use this identity

Answer 2

Use the substitution

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

then

cos^2(x) + 2cos^2(x) - 1 = 5/4

or

3cos^2(x) = 9/4

cos^2(x) = 3/4

\[\cos(x) = \pm \sqrt{3}/2\]

\[\cos^{-1}(\sqrt{3}/2) = \pi/3\]

you'll need to find the rest of the values of x over the given domain