Simplifying Trig Expressions: Step-by-Step Guide
Simplifying a trig expression means rewriting it in a shorter form that has the same value. You do this using identities (rules that are always true), like the Pythagorean, reciprocal, quotient, cofunction, and angle-sum identities. The goal is to use fewer different trig functions and to combine terms.
Math Racer skill guide · Trigonometry · Grades 11-12
How to solve it, step by step
- Change everything to sine and cosine. Replace tan, cot, sec, and csc with their sine and cosine forms. This shows common factors.\[\tan x = \frac{\sin x}{\cos x},\;\sec x = \frac{1}{\cos x}\]
- Combine fractions. Find common denominators, add or subtract fractions, and factor where you can.\[\frac{a}{c} + \frac{b}{c} = \frac{a + b}{c}\]
- Use the Pythagorean identities. Replace sin^2 + cos^2 with 1, or 1 - sin^2 with cos^2, and so on.\[\sin^2 x + \cos^2 x = 1\]
- Cancel and factor. Cancel common factors from the top and the bottom. Factor differences of squares.\[a^2 - b^2 = (a-b)(a+b)\]
- Change back if asked. If the problem wants the answer in a certain form (like one trig function), rewrite it that way.\[\frac{1}{\cos x} = \sec x\]
Worked example
Simplify (1 - cos^2 x) / sin x.
Step 1: The expression is already in sine and cosine.
Step 2: There are no fractions to combine.
Step 3: Use the Pythagorean identity: 1 - cos^2 x = sin^2 x.
Step 4: (sin^2 x) / (sin x) = sin x.
Step 5: The answer is sin x.
Check: At x = pi / 4, the original is (1 - 1/2) / (root 2 / 2) = (1/2) / (root 2 / 2) = 1 / root 2 = root 2 / 2. And sin(pi / 4) = root 2 / 2. They match.
Tips and common mistakes
- Changing to sine and cosine is almost always a good first step.
- Look for pieces that match a Pythagorean identity. They often simplify a lot.
- Factor before you cancel. You cannot cancel terms that are added.
- If you have 1 + sin x or 1 - cos x in a denominator, try multiplying by the conjugate.
- Watch for hidden differences of squares, like 1 - sin^2 x or sec^2 x - 1.
- Remember which values make the original or the simplified expression undefined.
- Check by trying a number like pi / 5 in both and comparing the decimals.
- If you are asked to prove an identity, work on each side separately until they match. Do not move things across the equals sign.