Proving identities is a major part of any trigonometry course. This involves getting one of the equation to be identical to the other side. Try using one of the following strategies to begin, and then use others if necessary to continue the verification process.
- First, start with what appears to be the more difficult side of the given identity.
- Try to transform this side so that it eventually matches identically to the other side.
- Don’t hesitate to start over and work with the other side of the identity if you are having trouble.
1. Look for ways to use known identities such as the reciprocal identities, quotient identities, and even/odd properties. If the identity includes a squared trigonometric expression. try using a variation of a Pythagorean identity.
2. Try rewriting each trigonometric expression in terms of sines and cosines.
3. Factor out a greatest common factor and use algebraic factoring techniques such as factoring the difference of two squares or the sum/difference of two cubes.
4. If a single term appears in the denominator of a quotient, try separating the quotient in two or more quotients:
$$\frac{A+B}{C} = \frac{A}{C}+\frac{B}{C}$$
5. If there are two or more fractional expressions, try combining the expressions using a common denominator:
$$\frac{A}{C}+\frac{B}{D} = \frac{AD+BC}{CD}$$
6. If the numerator or denominator of one or more quotients contains an expression of the form $B + \sqrt{C}$, try multiplying the numerator and denominator by its conjugate $B – \sqrt{C}$:
$$\frac{A}{B+\sqrt{C}}$$ $$= \frac{A}{B+ \sqrt{C}} \cdot \color{red} \frac{B- \sqrt{C}}{B-\sqrt{C}}$$ $$=\frac{A(B-\sqrt{C})}{B^2-C}$$
The idea here is that multiplying by the denominator’s conjugate, you force the denominator to become radical-less. This can lead to further manipulation of the terms.