Simplifying trigonometric expressions

If you get stuck trying to establish a trigonometric identity or solve a trigonometric equation, one strategy is to systematically simplify both sides (or to subtract them and simplify the result). By ‘simplify’ we really mean to put into a standard form, which is usually simpler than what you started with but not always. (Compare the process of rationalizing a denominator or expanding a factored polynomial.)

Here is a process for simplifying expressions with trigonometric operations:

Rewrite all operations in terms of sines and cosines alone.
If any expressions inside the trigonometric operations are more complicated than a single variable, then apply sum-angle and half-angle formulas as much as possible.
Simplify the result algebraically. That is, if you substitute x for cos θ and y for sin θ (and similarly for any other angle variables), you would simplify this non-trigonometric expression.
If a cosine appears with an exponent of at least 2, then change cos² θ to 1 − sin² θ (and similarly for any other angle variables) as much as possible; continue to simplify algebraically.
If any cosines still appear in a denominator, think of them as square roots and sine-ize the denominator in the same way that you would rationalize a denominator with a square root. That is, if you substitute √x for cos θ (and similarly for any other angle variables), you would rationalize the denominators with this radical.
If this produces more cos², then change these to sin² again; continue to simplify algebraically.
When your expression is written entirely in terms of sines and cosines, with the cosines only in numerators and never with exponents, with the expressions inside the sines and cosines as simple as possible and with the entire expression simplified algebraically as much as possible, now it is simplified!

If two sides of an identity are simplified in this manner, then they will be the same; similarly, an expression equivalent to 0 will simply to 0 in this way.

Instead of changing cos² to sin² and sinizing denominators, you can also change sin² to cos² and cosinize denominators. Just be consistent within a given problem.

This approach can also be extended to inverse trigonometric operations, but we don't need that for this class.

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This web page was written in 2014 by Toby Bartels, last edited on 2014 January 21. Toby reserves no legal rights to it.

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