Prove identity then solve equation

4

1. Show that \(3 \tan 2 \theta + \tan \left( \theta + 45 ^ { \circ } \right) \equiv \frac { \tan ^ { 2 } \theta + 8 \tan \theta + 1 } { 1 - \tan ^ { 2 } \theta }\).
2. Hence solve the equation \(3 \tan 2 \theta + \tan \left( \theta + 45 ^ { \circ } \right) = 4\) for \(0 ^ { \circ } < \theta < 180 ^ { \circ }\).

2

1. Prove the identity $$\cos \left( x + 30 ^ { \circ } \right) + \sin \left( x + 60 ^ { \circ } \right) \equiv ( \sqrt { } 3 ) \cos x$$
2. Hence solve the equation $$\cos \left( x + 30 ^ { \circ } \right) + \sin \left( x + 60 ^ { \circ } \right) = 1$$ for \(0 ^ { \circ } < x < 90 ^ { \circ }\).

8

1. Prove the identity $$\sin \left( x - 30 ^ { \circ } \right) + \cos \left( x - 60 ^ { \circ } \right) \equiv ( \sqrt { } 3 ) \sin x$$
2. Hence solve the equation $$\sin \left( x - 30 ^ { \circ } \right) + \cos \left( x - 60 ^ { \circ } \right) = \frac { 1 } { 2 } \sec x$$ for \(0 ^ { \circ } < x < 360 ^ { \circ }\).

8

1. Prove the identity $$\frac { 1 } { \sin \left( x - 60 ^ { \circ } \right) + \cos \left( x - 30 ^ { \circ } \right) } \equiv \operatorname { cosec } x$$
2. Hence solve the equation $$\frac { 2 } { \sin \left( x - 60 ^ { \circ } \right) + \cos \left( x - 30 ^ { \circ } \right) } = 3 \cot ^ { 2 } x - 2$$ for \(0 ^ { \circ } < x < 360 ^ { \circ }\).

6

1. Prove the identity $$\cos 4 \theta + 4 \cos 2 \theta \equiv 8 \cos ^ { 4 } \theta - 3$$
2. Hence solve the equation $$\cos 4 \theta + 4 \cos 2 \theta = 2$$ for \(0 ^ { \circ } \leqslant \theta \leqslant 360 ^ { \circ }\).

5

1. Prove the identity \(\cos 4 \theta - 4 \cos 2 \theta \equiv 8 \sin ^ { 4 } \theta - 3\).
2. Hence solve the equation $$\cos 4 \theta = 4 \cos 2 \theta + 3$$ for \(0 ^ { \circ } \leqslant \theta \leqslant 360 ^ { \circ }\).

4

1. Prove the identity $$\tan \left( x + 45 ^ { \circ } \right) - \tan \left( 45 ^ { \circ } - x \right) \equiv 2 \tan 2 x .$$
2. Hence solve the equation $$\tan \left( x + 45 ^ { \circ } \right) - \tan \left( 45 ^ { \circ } - x \right) = 2 ,$$ for \(0 ^ { \circ } \leqslant x \leqslant 180 ^ { \circ }\).

7

1. By expressing \(3 \theta\) as \(2 \theta + \theta\), prove the identity \(\cos 3 \theta \equiv 4 \cos ^ { 3 } \theta - 3 \cos \theta\).
2. Hence solve the equation $$\cos 3 \theta + \cos \theta \cos 2 \theta = \cos ^ { 2 } \theta$$ for \(0 ^ { \circ } \leqslant \theta \leqslant 180 ^ { \circ }\).

6. (i) Without using a calculator, find the exact value of $$\left( \sin 22.5 ^ { \circ } + \cos 22.5 ^ { \circ } \right) ^ { 2 }$$ You must show each stage of your working.
(ii) (a) Show that \(\cos 2 \theta + \sin \theta = 1\) may be written in the form $$k \sin ^ { 2 } \theta - \sin \theta = 0 , \text { stating the value of } k$$ (b) Hence solve, for \(0 \leqslant \theta < 360 ^ { \circ }\), the equation $$\cos 2 \theta + \sin \theta = 1$$

7. (i) Prove that, for \(\cos x \neq 0\), $$\sin 2 x - \tan x \equiv \tan x \cos 2 x$$ (ii) Hence, or otherwise, solve the equation $$\sin 2 x - \tan x = 2 \cos 2 x$$ for \(x\) in the interval \(0 \leq x \leq 180 ^ { \circ }\).

3. (i) Use the identity for \(\sin ( A + B )\) to show that $$\sin 3 x \equiv 3 \sin x - 4 \sin ^ { 3 } x$$ (ii) Hence find, in terms of \(\pi\), the solutions of the equation $$\sin 3 x - \sin x = 0$$ for \(x\) in the interval \(0 \leq x < 2 \pi\).

1. (i) Use the identity for \(\cos ( A + B )\) to prove that

$$\cos 2 x \equiv 2 \cos ^ { 2 } x - 1$$ (ii) Prove that, for \(\cos x \neq 0\), $$2 \cos x - \sec x \equiv \sec x \cos 2 x$$ (iii) Hence, or otherwise, find the values of \(x\) in the interval \(0 \leq x \leq 180 ^ { \circ }\) for which $$2 \cos x - \sec x \equiv 2 \cos 2 x$$

2 Show that \(\cot 2 \theta = \frac { 1 - \tan ^ { 2 } \theta } { 2 \tan \theta }\).
Hence solve the equation $$\cot 2 \theta = 1 + \tan \theta \quad \text { for } 0 ^ { \circ } < \theta < 360 ^ { \circ }$$

10. a. Show that $$\sin 3 A \equiv 3 \sin A - 4 \sin ^ { 3 } A$$ b. Hence solve, for \(- \frac { \pi } { 2 } \leq \theta \leq \frac { \pi } { 2 }\) the equation $$1 + \sin 3 \theta = \cos ^ { 2 } \theta$$

1. In this question you should show all stages of your working.

Solutions relying entirely on calculator technology are not acceptable.

1. Given that \(1 + \cos 2 \theta + \sin 2 \theta \neq 0\) prove that $$\frac { 1 - \cos 2 \theta + \sin 2 \theta } { 1 + \cos 2 \theta + \sin 2 \theta } \equiv \tan \theta$$
2. Hence solve, for \(0 < x < 180 ^ { \circ }\) $$\frac { 1 - \cos 4 x + \sin 4 x } { 1 + \cos 4 x + \sin 4 x } = 3 \sin 2 x$$ giving your answers to one decimal place where appropriate.

1. (a) Prove that

$$1 - \cos 2 \theta \equiv \tan \theta \sin 2 \theta , \quad \theta \neq \frac { ( 2 n + 1 ) \pi } { 2 } , \quad n \in \mathbb { Z }$$ (b) Hence solve, for \(- \frac { \pi } { 2 } < x < \frac { \pi } { 2 }\), the equation $$\left( \sec ^ { 2 } x - 5 \right) ( 1 - \cos 2 x ) = 3 \tan ^ { 2 } x \sin 2 x$$ Give any non-exact answer to 3 decimal places where appropriate.

1. In this question you must show all stages of your working.

Solutions relying entirely on calculator technology are not acceptable.

1. Show that $$\cos 3 A \equiv 4 \cos ^ { 3 } A - 3 \cos A$$
2. Hence solve, for \(- 90 ^ { \circ } \leqslant x \leqslant 180 ^ { \circ }\), the equation $$1 - \cos 3 x = \sin ^ { 2 } x$$

a) Given that \(\alpha\) and \(\beta\) are two angles such that \(\tan \alpha = 2 \cot \beta\), show that $$\tan ( \alpha + \beta ) = - ( \tan \alpha + \tan \beta )$$ b) Find all values of \(\theta\) in the range \(0 ^ { \circ } \leqslant \theta \leqslant 360 ^ { \circ }\) satisfying the equation $$4 \tan \theta = 3 \sec ^ { 2 } \theta - 7$$