Cotangent Identity
Cotangent as the ratio of cosine to sine.
Formula
Variables
Example
cot 45° = cos45°/sin45° = 1.
How it works
The cotangent of an angle is defined as the ratio of the adjacent side to the opposite side in a right triangle, and that same ratio can be rewritten using sine and cosine because costheta = adjacent/hypotenuse and sintheta = opposite/hypotenuse. Dividing cosine by sine cancels the hypotenuse, leaving adjacent/opposite, which is exactly cottheta. This identity is useful because it connects cotangent to the two primary trig functions and lets you simplify expressions or rewrite everything in terms of sine and cosine.
When to use it
Use this identity when simplifying trigonometric expressions, especially if you want to convert cotangent into sine and cosine. It is also helpful in solving equations, proving trig identities, and working with derivatives or integrals involving trig ratios.
Worked example, step by step
Find cot 30 b0 using the identity. cos 30 b0 = 3/2 sin 30 b0 = 1/2 cot 30 b0 = cos 30 b0 / sin 30 b0 cot 30 b0 = (3/2) / (1/2) Cot 30 b0 = 3
Common mistakes to avoid
- Swapping sine and cosine, which gives tan b8 instead of cot b8
- Forgetting that cotangent is undefined when sin b8 = 0
- Using degrees and radians inconsistently when evaluating the angle
Did You Know?
Cotangent is the reciprocal of tangent — cosine over sine.
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