Standard +0.3 This is a standard C3 trigonometric identities question with three routine parts: (i) derive a double-angle formula using a given identity (straightforward substitution), (ii) prove a standard half-angle identity using part (i), and (iii) solve an equation by substituting the proven identities. All techniques are textbook exercises requiring methodical application rather than insight, making it slightly easier than average.
7. (i) Use the identity
$$\cos ( A + B ) \equiv \cos A \cos B - \sin A \sin B$$
to prove that
$$\cos x \equiv 1 - 2 \sin ^ { 2 } \frac { x } { 2 }$$
(ii) Prove that, for \(\sin x \neq 0\),
$$\frac { 1 - \cos x } { \sin x } \equiv \tan \frac { x } { 2 }$$
(iii) Find the values of \(x\) in the interval \(0 \leq x \leq 360 ^ { \circ }\) for which
$$\frac { 1 - \cos x } { \sin x } = 2 \sec ^ { 2 } \frac { x } { 2 } - 5$$
giving your answers to 1 decimal place where appropriate.
7. (i) Use the identity
$$\cos ( A + B ) \equiv \cos A \cos B - \sin A \sin B$$
to prove that
$$\cos x \equiv 1 - 2 \sin ^ { 2 } \frac { x } { 2 }$$
(ii) Prove that, for $\sin x \neq 0$,
$$\frac { 1 - \cos x } { \sin x } \equiv \tan \frac { x } { 2 }$$
(iii) Find the values of $x$ in the interval $0 \leq x \leq 360 ^ { \circ }$ for which
$$\frac { 1 - \cos x } { \sin x } = 2 \sec ^ { 2 } \frac { x } { 2 } - 5$$
giving your answers to 1 decimal place where appropriate.\\
\hfill \mbox{\textit{OCR C3 Q7 [11]}}