Q BankQuestion BankDocsDocuments

3.3.2—Trig identities

Syllabus
9709–2028–2029
Objective
3.3.2
Level
A2

Select the identity family that exposes the required form

1+ an^2A=\sec^2A,\qquad 1+\cot^2A=\cosec^2A.Usethesetoexchangeareciprocalsquareforatangent/cotangentsquare.Use these to exchange a reciprocal square for a tangent/cotangent square.

\sin(A\pm B)=\sin A\cos B\pm\cos A\sin B,\cos(A\pm B)=\cos A\cos B\mp\sin A\sin B, an(A\pm B)= rac{ an A\pm an B}{1\mp an A an B}.

\sin2A=2\sin A\cos A,\quad \cos2A=\cos^2A-\sin^2A=1-2\sin^2A=2\cos^2A-1, an2A= rac{2 an A}{1- an^2A}.

Write asin⁡heta+bcos⁡heta=Rsin⁡(heta+α)a\sin heta+b\cos heta=R\sin( heta+\alpha) by matching Rcos⁡α=aR\cos\alpha=a and Rsin⁡α=bR\sin\alpha=b. Thus R=a2+b2R=\sqrt{a^2+b^2} and choose the quadrant of α\alpha from both signs. A cosine form is equally valid when coefficient matching is adjusted.

For exact values, expand a compound angle built from known angles. For equations, rewrite to one function or RR-form, use its range to decide existence, find all solutions in the stated interval, and check any denominator restrictions.

The sign in the cosine compound formula is opposite the sign between the angles. Identity manipulation does not remove excluded values introduced by dividing by sine, cosine or another expression.

ConceptA-Level CAIE Mathematics A2