Simplifying trigonometric expressions is a matter of understanding the circles and triangles upon which trigonometry is based. While much of the simplification can be done geometrically, knowledge of trigonometric identities will allow an algebraic solution.
- Know your angles
Every time you see sin x or cos x, know that x can be replaced with x-2? or x+2? (if in radians) or x-360 or x+360 (if in degrees) with no change in value. Sine and cosine are periodic functions, meaning that their values repeat every 2? or 360 degrees. Further, when you see tan x or cot x, it is periodic with repeating values every ? radians or 180 degrees, so x can be replaced with x+? or x-?, for example, sin 7? = sin 5? = sin 3? = sin ? = 0.
- Use Pythagorean identities
When some terms are difficult to simplify, using these identities can yield a different result that is more manageable : sin^2 x + cos^2 x = 1 tan^2 x + 1 = sec^2 x cot^2 x + 1 = csc^2 x Keep in mind that these equations can also be manipulated to fit what you need. What does sec^2 x - 1 equal?
- Discover your own identities
The reference for this article includes many more identities that can be substituted. The main aspect in simplifying is to keep an open mind when looking at each term. Knowing that each term can be replaced by terms with different angles or different expressions altogether leaves many options open to mold the expression into something that fits better.