The three Pythagorean identities
Starting from sin²θ + cos²θ = 1 and dividing through by cos²θ or sin²θ gives three related identities that hold for every angle where the functions are defined.
Key formula /
sin²θ + cos²θ = 1; 1 + tan²θ = sec²θ; 1 + cot²θ = cosec²θ. Quotient: tan θ = sin θ / cos θ, cot θ = cos θ / sin θ.
Using an identity
Identities are used in two ways: to simplify an expression to a single ratio, and to evaluate a second ratio from a known one. When taking a square root, choose the sign from the quadrant.
These identities are also the main tools for proving harder statements. The usual method is to work on the more complicated side, rewrite every term using only sine and cosine, and simplify until the two sides agree. Because an identity holds for every angle where the functions are defined, you can always check your working by substituting a convenient value such as θ = 30° and confirming both sides give the same number.
Worked example
Given sin θ = 3/5 and θ acute, find cos θ and tan θ. From sin²θ + cos²θ = 1, cos²θ = 1 − 9/25 = 16/25, so cos θ = 4/5 (positive, acute). Then tan θ = (3/5) ÷ (4/5) = 3/4.
Simplify (1 − cos²θ)/sin θ. The numerator is sin²θ, so the expression is sin²θ / sin θ = sin θ.
Remember
- sec θ = 1/cos θ, cosec θ = 1/sin θ, cot θ = 1/tan θ.
- 1 − cos²θ = sin²θ and 1 − sin²θ = cos²θ.
- Decide the sign of a root from the quadrant.