Product to sum using compound angles - Addition & Double Angle Formulae

Edexcel C3 Q5

11 marks Moderate -0.5

5. (a) Using the formulae $$\begin{gathered} \sin ( A \pm B ) = \sin A \cos B \pm \cos A \sin B \\ \cos ( A \pm B ) = \cos A \cos B \mp \sin A \sin B \end{gathered}$$ show that

$\sin ( A + B ) - \sin ( A - B ) = 2 \cos A \sin B$,
$\cos ( A - B ) - \cos ( A + B ) = 2 \sin A \sin B$.
(b) Use the above results to show that $$\frac { \sin ( A + B ) - \sin ( A - B ) } { \cos ( A - B ) - \cos ( A + B ) } = \cot A$$ Using the result of part (b) and the exact values of $\sin 60 ^ { \circ }$ and $\cos 60 ^ { \circ }$,
(c) find an exact value for $\cot 75 ^ { \circ }$ in its simplest form.

5. continued Leave blank

Edexcel C3 Q3

9 marks Standard +0.3

3. (a) Use the identities for $\sin ( A + B )$ and $\sin ( A - B )$ to prove that $$\sin P + \sin Q \equiv 2 \sin \frac { P + Q } { 2 } \cos \frac { P - Q } { 2 } \text {. }$$ (b) Find, in terms of $\pi$, the solutions of the equation $$\sin 5 x + \sin x = 0$$ for $x$ in the interval $0 \leq x < \pi$.

Edexcel C3 Q4

9 marks Challenging +1.2

4. (a) Use the identities for ( $\sin A + \sin B$ ) and ( $\cos A + \cos B$ ) to prove that $$\frac { \sin 2 x + \sin 2 y } { \cos 2 x + \cos 2 y } \equiv \tan ( x + y ) .$$ (b) Hence, show that $$\tan 52.5 ^ { \circ } = \sqrt { 6 } - \sqrt { 3 } - \sqrt { 2 } + 2 .$$

Edexcel C3 Q2

9 marks Standard +0.3

2. (a) Use the identities for $\cos ( A + B )$ and $\cos ( A - B )$ to prove that $$2 \cos A \cos B \equiv \cos ( A + B ) + \cos ( A - B ) .$$ (b) Hence, or otherwise, find in terms of $\pi$ the solutions of the equation $$2 \cos \left( x + \frac { \pi } { 2 } \right) = \sec \left( x + \frac { \pi } { 6 } \right) ,$$ for $x$ in the interval $0 \leq x \leq \pi$.

Edexcel C3 Q6

11 marks Standard +0.8

(a) Use the identities for $\cos ( A + B )$ and $\cos ( A - B )$ to prove that

$$\cos P - \cos Q \equiv - 2 \sin \frac { P + Q } { 2 } \sin \frac { P - Q } { 2 }$$ (b) Hence find all solutions in the interval $0 \leq x < 180$ to the equation $$\cos 5 x ^ { \circ } + \sin 3 x ^ { \circ } - \cos x ^ { \circ } = 0$$

Edexcel C2 Q4

9 marks Moderate -0.3

Write down formulae for sin (A + B) and sin (A - B). Using X = A + B and Y = A - B, prove that $$\sin X + \sin Y = 2 \sin \frac{X + Y}{2} \cos \frac{X - Y}{2}.$$ [4 marks]
Hence, or otherwise, solve, for 0 ≤ θ < 360, $$\sin 40° + \sin 20° = 0.$$ [5 marks]

SPS SPS FM Pure 2021 May Q8

8 marks Hard +2.3

Let $C = \sum_{r=0}^{20} \binom{20}{r} \cos(r\theta)$. Show that $C = 2^{20} \cos^{20}\left(\frac{1}{2}\theta\right) \cos(10\theta)$. [8]