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Use the double-angle formula for cosine to write in terms of
Establishing identities using the double-angle formulas is performed using the same steps we used to derive the sum and difference formulas. Choose the more complicated side of the equation and rewrite it until it matches the other side.
Establish the following identity using double-angle formulas:
We will work on the right side of the equal sign and rewrite the expression until it matches the left side.
Verify the identity:
In this case, we will work with the left side of the equation and simplify or rewrite until it equals the right side of the equation.
The double-angle formulas can be used to derive the reduction formulas , which are formulas we can use to reduce the power of a given expression involving even powers of sine or cosine. They allow us to rewrite the even powers of sine or cosine in terms of the first power of cosine. These formulas are especially important in higher-level math courses, calculus in particular. Also called the power-reducing formulas, three identities are included and are easily derived from the double-angle formulas.
We can use two of the three double-angle formulas for cosine to derive the reduction formulas for sine and cosine. Let’s begin with Solve for
Next, we use the formula Solve for
The last reduction formula is derived by writing tangent in terms of sine and cosine:
The reduction formulas are summarized as follows:
Write an equivalent expression for that does not involve any powers of sine or cosine greater than 1.
We will apply the reduction formula for cosine twice.
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