Pythagorean identities and proofs

An identity holds for every angle where both sides are defined. Quotient identities tan A= sin Acos A A= cos Asin A Require nonzero denominators. The form cot A= 1tan A also requires tan A to exist and be nonzero. Pythagorean identities sin2A+cos2A=1 1+tan2A=sec2A 1+cot2A=cosec2A The last two follow by dividing by cosine squared or sine squared respectively. Use the same angle throughout. **Correct use** | **Not an identity** sin2 3theta+cos2 3theta=1 | sintheta+costheta=1 sec2 theta2=1+tan2 theta2 | sin4theta+cos4theta=1 Proving an identity Start with one side, express reciprocal ratios in sine and cosine, then combine fractions or factorise until the other side is obtained. **Move** | **Working** Express in sin and cos | cot Asin A = cos Asin A x sin A =cos A Pythagorean swap after factorising | (1-cos A)(1+cos A) =1-cos2 A =sin2 A Combine into one fraction | 1cos2 A-1 = 1-cos2 Acos2 A = sin2 Acos2 A =tan2 A Do not assume the identity you are proving. With r2=x2+y2, division by r2 gives sin2theta+cos2theta=1. Division by cos2theta or sin2theta, when nonzero, gives the other two identities.