Power-Reducing Formulas
Rewrite squared functions using double angles
8.17 List the power-reducing formulas for cos2 θ and sin2θ.
Power-reducing formulas not only remove the exponents from the trigonometric expression, they also rewrite the expression entirely in terms of cosine.
You end up with cosines of double angles.
Expand the right side of the power-reducing formula by applying the double-angle formula cos 2θ = 1 – 2 sin2θ.
The final statement sin2θ = sin2θ is obviously true; therefore, the original statement is verified.
8.19 Apply a power-reducing formula to express cos4θ as a sum of cosine functions raised to the first power.
Express cos4θ as (cos2 θ)2 and apply the power-reducing formula to rewrite cos2θ.
To rewrite cos2 2θ using the power-reducing formula 
replace x with 2θ
Apply the power-reducing formula once again to eliminate the exponent from the expression.
Add the constants to simplify the expression: 1/4 + 1/8 = 3/8.
The solution (1/8)(3 + 4 cos 2θ + cos 4θ) is also correct.
8.20 Apply a power-reducing formula to express cos2 x + sin4 x as a sum that includes cosine functions raised to the first power.
Express sin4 x as the product (sin2 x)(sin2 x) and apply the power-reducing formulas.
Once again, apply the power-reducing formula 
, this time replacing θ with 2x.
HANDY TIP: To check your answer for this (or any other) problem that asks you to simplify, graph the original expression and the simplified version as two separate equations on your graphing calculator. If the graphs overlap, you simplified correctly.
8.21 Verify the statement: (sin2θ)(cos2 θ) = 
(sin2 2θ).
Rewrite the factors on the left side of the equation, applying power-reducing formulas.
Multiply the entire equation by 4 to eliminate the fractions.
If x = 2θ, then the final statement is equivalent to the Pythagorean identity cos2 x + sin2 x = 1. Because that statement is true, the original statement is verified.