CAIE A-Level Mathematics 1.5.4 Trig identities

$\frac{(\cos \theta-\sin \theta) \sin \theta+(\cos \theta+\sin \theta)(1-\cos \theta)}{(\cos \theta+\sin \theta)(\cos \theta-\sin \theta)}$

M1

Correct common denominator and correct products in the numerator and no missing terms. Correct factors in the
denominator can be implied by $\cos ^{2} \theta-\sin ^{2} \theta$. Condone
brackets missing if recovered.

$=\frac{\cos \theta \sin \theta-\sin ^{2} \theta+\cos \theta-\cos ^{2} \theta+\sin \theta-\sin \theta \cos \theta}{\cos ^{2} \theta-\sin ^{2} \theta}$

A1

$=\frac{\sin \theta+\cos \theta-\cos ^{2} \theta-\sin ^{2} \theta}{\cos ^{2} \theta-\sin ^{2} \theta}=\frac{\cos \theta+\sin \theta-1}{1-2 \sin ^{2} \theta}$

A1

AG

Clear evidence of using $\sin ^{2} \theta+\cos ^{2} \theta=1$ in either the numerator or denominator. Condone c, s and/or omission of $\theta$.

Working from both sides of the identity and correctly arriving at the same expression can score M1A1. A final statement is then required for the A1.