CAIE A-Level Mathematics A2 3.3.2 Trig Identities Questions

Factorise LHS using difference of 2 squares

*M1

$\left(\left(\cos ^{2} \theta-\sin ^{2} \theta\right)\left(\cos ^{2} \theta+\sin ^{2} \theta\right)\right)$

Simplify
$\cos ^{2} \theta+\sin ^{2} \theta=1$ must be seen or implied, e.g.
$\left(\cos ^{2} \theta-\sin ^{2} \theta\right)\left(\cos ^{2} \theta+\sin ^{2} \theta\right)=\left(\cos ^{2} \theta-\sin ^{2} \theta\right)$.

Obtain $\cos ^{4} \theta-\sin ^{4} \theta \equiv \cos 2 \theta$ from correct working
AG

Alternative Method for Question 7(a)

Use of correct rearrangements of double angle formulae

(*M1)

E.g. $\left(\frac{1+\cos 2 \theta}{2}\right)^{2}-\left(\frac{1-\cos 2 \theta}{2}\right)^{2}$

Only condone $\left(\frac{1+\cos 2 \theta}{2}\right)^{2}-\left(\frac{\cos 2 \theta-1}{2}\right)^{2}$ if correct expression for $\sin ^{2} \theta$ seen.

Expand and simplify
Collect like terms. Condone recovery from missing brackets.

Obtain $\cos ^{4} \theta-\sin ^{4} \theta \equiv \cos 2 \theta$ from correct working
AG

Alternative Method 2 for Question 7(a)

Correct use of Pythagoras

(*M1)

E.g. $\left(1-\sin ^{2} \theta\right)^{2}-\sin ^{4} \theta$ or $\cos ^{2} \theta\left(1-\sin ^{2} \theta\right)-\sin ^{2} \theta\left(1-\cos ^{2} \theta\right)$

Expand and simplify
Condone recovery from missing brackets.

Obtain $\cos ^{4} \theta-\sin ^{4} \theta \equiv \cos 2 \theta$ from correct working
AG