Proof of the reciprocal identities Proof of the tangent and cotangent identities Proof of the Pythagorean identities The proof of each of those follows from the definitions of the trigonometric functions, Topic 15. Proof of the reciprocal relations By definition:
Therefore, sin θ is the reciprocal of csc θ:
where 1-over any quantity is the symbol for its reciprocal; Lesson 5 of Algebra. Similarly for the remaining functions. Proof of the tangent and cotangent identities To prove:
Therefore, on dividing both numerator and denominator by
Those are the two identities. Proof of the Pythagorean identities To prove:
Therefore, on dividing both sides by
That is, according to the definitions, cos Apart from the order of the terms, this is the first Pythagorean identity, a). To derive b), divide line (1) by Or, we can derive both b) and c) from a) by dividing it first by cos That is, 1 + tan And if we divide a) by sin That is, 1 + cot The three Pythagorean identities are thus equivalent to one another.
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