Product to sum using compound angles

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

1. \(\sin ( A + B ) - \sin ( A - B ) = 2 \cos A \sin B\),
2. \(\cos ( A - B ) - \cos ( A + B ) = 2 \sin A \sin B\).
   (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.
   

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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 \(0 \leq x < \pi\).

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 $$\tan 52.5 ^ { \circ } = \sqrt { 6 } - \sqrt { 3 } - \sqrt { 2 } + 2 .$$

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 \(0 \leq x \leq \pi\).

1. (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 $$\cos 5 x ^ { \circ } + \sin 3 x ^ { \circ } - \cos x ^ { \circ } = 0$$ Turn over

5. (a) Show that \(\sin \theta - \sin 3 \theta\) can be expressed in the form \(a \cos b \theta \sin \theta\), where \(a , b\) are integers whose values are to be determined.
(b) Find the mean value of \(y = 2 \cos 2 \theta \sin \theta + 7\) between \(\theta = 1\) and \(\theta = 3\), giving your answer correct to two decimal places.