Double angle with reciprocal functions

1. (a) Prove that

$$2 \operatorname { cosec } ^ { 2 } 2 \theta ( 1 - \cos 2 \theta ) \equiv 1 + \tan ^ { 2 } \theta$$ (b) Hence solve for \(0 < x < 360 ^ { \circ }\), where \(x \neq ( 90 n ) ^ { \circ } , n \in \mathbb { N }\), the equation $$2 \operatorname { cosec } ^ { 2 } 2 x ( 1 - \cos 2 x ) = 4 + 3 \sec x$$ giving your answers to one decimal place.
(Solutions relying entirely on calculator technology are not acceptable.)

8. (a) Prove that $$\text { 2cosec } 2 A - \cot A \equiv \tan A , \quad A \neq \frac { n \pi } { 2 } , n \in \mathbb { Z }$$ (b) Hence solve, for \(0 \leqslant \theta \leqslant \frac { \pi } { 2 }\)

1. \(2 \operatorname { cosec } 4 \theta - \cot 2 \theta = \sqrt { } 3\)
2. \(\tan \theta + \cot \theta = 5\) Give your answers to 3 significant figures.

12. (a) Show that $$\cot x - \tan x \equiv 2 \cot 2 x , \quad x \neq 90 n ^ { \circ } , n \in \mathbb { Z }$$ (b) Hence, or otherwise, solve, for \(0 \leqslant \theta < 180 ^ { \circ }\) $$5 + \cot \left( \theta - 15 ^ { \circ } \right) - \tan \left( \theta - 15 ^ { \circ } \right) = 0$$ giving your answers to one decimal place.
[0pt] [Solutions based entirely on graphical or numerical methods are not acceptable.]

1. (a) Show that

$$\cot x - \cot 2 x \equiv \operatorname { cosec } 2 x , \quad x \neq \frac { n \pi } { 2 } , \quad n \in \mathbb { Z }$$ (b) Hence, or otherwise, solve for \(0 \leqslant \theta \leqslant \pi\) $$\operatorname { cosec } \left( 3 \theta + \frac { \pi } { 3 } \right) + \cot \left( 3 \theta + \frac { \pi } { 3 } \right) = \frac { 1 } { \sqrt { 3 } }$$ You must show your working.
(Solutions based entirely on graphical or numerical methods are not acceptable.)

7. (a) Show that

1. \(\frac { \cos 2 x } { \cos x + \sin x } \equiv \cos x - \sin x , \quad x \neq \left( n - \frac { 1 } { 4 } \right) \pi , n \in \mathbb { Z }\),
2. \(\frac { 1 } { 2 } ( \cos 2 x - \sin 2 x ) \equiv \cos ^ { 2 } x - \cos x \sin x - \frac { 1 } { 2 }\).
   (b) Hence, or otherwise, show that the equation $$\cos \theta \left( \frac { \cos 2 \theta } { \cos \theta + \sin \theta } \right) = \frac { 1 } { 2 }$$ can be written as $$\sin 2 \theta = \cos 2 \theta$$ (c) Solve, for \(0 \leqslant \theta < 2 \pi\), $$\sin 2 \theta = \cos 2 \theta$$ giving your answers in terms of \(\pi\).

1. (a) Express \(4 \operatorname { cosec } ^ { 2 } 2 \theta - \operatorname { cosec } ^ { 2 } \theta\) in terms of \(\sin \theta\) and \(\cos \theta\).
   (b) Hence show that

$$4 \operatorname { cosec } ^ { 2 } 2 \theta - \operatorname { cosec } ^ { 2 } \theta = \sec ^ { 2 } \theta$$ (c) Hence or otherwise solve, for \(0 < \theta < \pi\), $$4 \operatorname { cosec } ^ { 2 } 2 \theta - \operatorname { cosec } ^ { 2 } \theta = 4$$ giving your answers in terms of \(\pi\).

1. (i) Use an appropriate double angle formula to show that

$$\operatorname { cosec } 2 x = \lambda \operatorname { cosec } x \sec x$$ and state the value of the constant \(\lambda\).
(ii) Solve, for \(0 \leqslant \theta < 2 \pi\), the equation $$3 \sec ^ { 2 } \theta + 3 \sec \theta = 2 \tan ^ { 2 } \theta$$ You must show all your working. Give your answers in terms of \(\pi\).