Double angle with reciprocal functions

7

1. Prove that \(2 \sin \theta \operatorname { cosec } 2 \theta \equiv \sec \theta\).
2. Solve the equation \(\tan ^ { 2 } \theta + 7 \sin \theta \operatorname { cosec } 2 \theta = 8\) for \(- \pi < \theta < \pi\). \includegraphics[max width=\textwidth, alt={}, center]{76df3465-9617-4f2b-a8b7-f474b2817504-12_2725_37_136_2010}
3. Find \(\int 8 \sin ^ { 2 } \frac { 1 } { 2 } x \operatorname { cosec } ^ { 2 } x \mathrm {~d} x\).
   If you use the following page to complete the answer to any question, the question number must be clearly shown. \includegraphics[max width=\textwidth, alt={}, center]{76df3465-9617-4f2b-a8b7-f474b2817504-14_2715_35_143_2012}

6

1. Show that \(\operatorname { cosec } \theta \left( 3 \sin 2 \theta + 4 \sin ^ { 3 } \theta \right) \equiv 4 + 6 \cos \theta - 4 \cos ^ { 2 } \theta\).
2. Solve the equation $$\operatorname { cosec } \theta \left( 3 \sin 2 \theta + 4 \sin ^ { 3 } \theta \right) + 3 = 0$$ for \(- \pi < \theta < 0\).
3. Find \(\int \operatorname { cosec } \theta \left( 3 \sin 2 \theta + 4 \sin ^ { 3 } \theta \right) \mathrm { d } \theta\).

2 Solve the equation \(3 \sin 2 \theta \tan \theta = 2\) for \(0 ^ { \circ } < \theta < 180 ^ { \circ }\).

8

1. Prove that \(\sin ^ { 2 } 2 \theta \left( \operatorname { cosec } ^ { 2 } \theta - \sec ^ { 2 } \theta \right) \equiv 4 \cos 2 \theta\).
2. Hence
   1. solve for \(0 ^ { \circ } \leqslant \theta \leqslant 180 ^ { \circ }\) the equation \(\sin ^ { 2 } 2 \theta \left( \operatorname { cosec } ^ { 2 } \theta - \sec ^ { 2 } \theta \right) = 3\),
   2. find the exact value of \(\operatorname { cosec } ^ { 2 } 15 ^ { \circ } - \sec ^ { 2 } 15 ^ { \circ }\).

6

1. Prove that \(2 \operatorname { cosec } 2 \theta \tan \theta \equiv \sec ^ { 2 } \theta\).
2. Hence
   1. solve the equation \(2 \operatorname { cosec } 2 \theta \tan \theta = 5\) for \(0 < \theta < \pi\),
   2. find the exact value of \(\int _ { 0 } ^ { \frac { 1 } { 6 } \pi } 2 \operatorname { cosec } 4 x \tan 2 x \mathrm {~d} x\).

7

1. Show that \(2 \operatorname { cosec } ^ { 2 } 2 x ( 1 - \cos 2 x ) \equiv \sec ^ { 2 } x\).
2. Solve the equation \(2 \operatorname { cosec } ^ { 2 } 2 x ( 1 - \cos 2 x ) = \tan x + 21\) for \(0 < x < \pi\), giving your answers correct to 3 significant figures.
3. Find \(\int \left[ 2 \operatorname { cosec } ^ { 2 } ( 4 y + 2 ) - 2 \operatorname { cosec } ^ { 2 } ( 4 y + 2 ) \cos ( 4 y + 2 ) \right] \mathrm { d } y\).
   If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.

1 Prove the identity $$\cot \theta - \tan \theta \equiv 2 \cot 2 \theta$$

3

1. Prove the identity \(\operatorname { cosec } 2 \theta + \cot 2 \theta \equiv \cot \theta\).
2. Hence solve the equation \(\operatorname { cosec } 2 \theta + \cot 2 \theta = 2\), for \(0 ^ { \circ } \leqslant \theta \leqslant 360 ^ { \circ }\).

4 Solve the equation $$\operatorname { cosec } 2 \theta = \sec \theta + \cot \theta$$ giving all solutions in the interval \(0 ^ { \circ } < \theta < 360 ^ { \circ }\).

3 Solve the equation \(\cot 2 x + \cot x = 3\) for \(0 ^ { \circ } < x < 180 ^ { \circ }\).

1 Prove the identity \(\frac { \cot x - \tan x } { \cot x + \tan x } \equiv \cos 2 x\).

7

1. Show that \(2 \operatorname { cosec } 2 \theta \cot \theta \equiv \operatorname { cosec } ^ { 2 } \theta\).
2. Hence show that \(\operatorname { cosec } ^ { 2 } 15 ^ { \circ } \tan 15 ^ { \circ } = 4\).
3. Solve the equation \(2 \operatorname { cosec } \phi \cot \frac { 1 } { 2 } \phi + \operatorname { cosec } \frac { 1 } { 2 } \phi = 12\) for \(- 360 ^ { \circ } < \phi < 360 ^ { \circ }\). Show all necessary working.
   If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.

8

1. Show that \(\sin 2 x \cot x \equiv 2 \cos ^ { 2 } x\).
2. Using the identity in part (i),
   1. find the least possible value of $$3 \sin 2 x \cot x + 5 \cos 2 x + 8$$ as \(x\) varies,
   2. find the exact value of \(\int _ { \frac { 1 } { 8 } \pi } ^ { \frac { 1 } { 6 } \pi } \operatorname { cosec } 4 x \tan 2 x \mathrm {~d} x\).

2

1. Given that \(\tan 2 \theta \cot \theta = 8\), show that \(\tan ^ { 2 } \theta = \frac { 3 } { 4 }\).
2. Hence solve the equation \(\tan 2 \theta \cot \theta = 8\) for \(0 ^ { \circ } < \theta < 180 ^ { \circ }\).

6

1. Show that \(\frac { \cos 2 \theta } { 1 + \cos 2 \theta } \equiv 1 - \frac { 1 } { 2 } \sec ^ { 2 } \theta\).
2. Solve the equation \(\frac { \cos 2 \alpha } { 1 + \cos 2 \alpha } = 13 + 5 \tan \alpha\) for \(0 < \alpha < \pi\).
3. Find the exact value of \(\int _ { - \frac { 1 } { 2 } \pi } ^ { \frac { 1 } { 2 } \pi } \frac { \cos x } { 1 + \cos x } \mathrm {~d} x\).

3 Solve the equation \(2 \cot 2 x + 3 \cot x = 5\), for \(0 ^ { \circ } < x < 180 ^ { \circ }\).

5

1. Show that \(\frac { \cos 3 x } { \sin x } + \frac { \sin 3 x } { \cos x } = 2 \cot 2 x\).
2. Hence solve the equation \(\frac { \cos 3 x } { \sin x } + \frac { \sin 3 x } { \cos x } = 4\), for \(0 < x < \pi\).

8

1. By first expanding \(\left( \cos ^ { 2 } \theta + \sin ^ { 2 } \theta \right) ^ { 2 }\), show that $$\cos ^ { 4 } \theta + \sin ^ { 4 } \theta \equiv 1 - \frac { 1 } { 2 } \sin ^ { 2 } 2 \theta .$$
2. Hence solve the equation $$\cos ^ { 4 } \theta + \sin ^ { 4 } \theta = \frac { 5 } { 9 } ,$$ for \(0 ^ { \circ } < \theta < 180 ^ { \circ }\).

6

1. Prove the identity \(\cos 4 \theta + 4 \cos 2 \theta + 3 \equiv 8 \cos ^ { 4 } \theta\).
2. Hence solve the equation \(\cos 4 \theta + 4 \cos 2 \theta = 4\) for \(0 ^ { \circ } \leqslant \theta \leqslant 180 ^ { \circ }\).

6

1. Show that the equation \(\cot ^ { 2 } \theta + 2 \cos 2 \theta = 4\) can be written in the form $$4 \sin ^ { 4 } \theta + 3 \sin ^ { 2 } \theta - 1 = 0$$
2. Hence solve the equation \(\cot ^ { 2 } \theta + 2 \cos 2 \theta = 4\), for \(0 ^ { \circ } < \theta < 360 ^ { \circ }\).

4

1. Show that \(\sec ^ { 4 } \theta - \tan ^ { 4 } \theta \equiv 1 + 2 \tan ^ { 2 } \theta\). \includegraphics[max width=\textwidth, alt={}, center]{656df2a8-fc4d-49f3-a649-746103b4576e-07_2723_35_101_20}
2. Hence or otherwise solve the equation \(\sec ^ { 4 } 2 \alpha - \tan ^ { 4 } 2 \alpha = 2 \tan ^ { 2 } 2 \alpha \sec ^ { 2 } 2 \alpha\) for \(0 ^ { \circ } < \alpha < 180 ^ { \circ }\). [5]

5

1. Show that \(\cos ^ { 4 } \theta - \sin ^ { 4 } \theta - 4 \sin ^ { 2 } \theta \cos ^ { 2 } \theta \equiv \cos ^ { 2 } 2 \theta + \cos 2 \theta - 1\).
2. Solve the equation \(\cos ^ { 4 } \alpha - \sin ^ { 4 } \alpha = 4 \sin ^ { 2 } \alpha \cos ^ { 2 } \alpha\) for \(0 ^ { \circ } \leqslant \alpha \leqslant 180 ^ { \circ }\).

7. (a) Prove that $$\frac { \sin 2 x } { \cos x } + \frac { \cos 2 x } { \sin x } \equiv \operatorname { cosec } x \quad x \neq \frac { n \pi } { 2 } n \in \mathbb { Z }$$ (b) Hence solve, for \(- \frac { \pi } { 2 } < \theta < \frac { \pi } { 2 }\) $$7 + \frac { \sin 4 \theta } { \cos 2 \theta } + \frac { \cos 4 \theta } { \sin 2 \theta } = 3 \cot ^ { 2 } 2 \theta$$ giving your answers in radians to 3 significant figures where appropriate.

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

Solutions relying entirely on calculator technology are not acceptable.

1. Prove that $$\cot ^ { 2 } x - \tan ^ { 2 } x \equiv 4 \cot 2 x \operatorname { cosec } 2 x \quad x \neq \frac { n \pi } { 2 } \quad n \in \mathbb { Z }$$
2. Hence solve, for \(- \frac { \pi } { 2 } < \theta < \frac { \pi } { 2 }\) $$4 \cot 2 \theta \operatorname { cosec } 2 \theta = 2 \tan ^ { 2 } \theta$$ giving your answers to 2 decimal places.

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

Solutions relying entirely on calculator technology are not acceptable.

1. Show that $$\frac { \cos 2 x } { \sin x } + \frac { \sin 2 x } { \cos x } \equiv \operatorname { cosec } x \quad x \neq \frac { n \pi } { 2 } \quad n \in \mathbb { Z }$$
2. Hence solve, for \(0 < \theta < \frac { \pi } { 2 }\) $$\left( \frac { \cos 2 \theta } { \sin \theta } + \frac { \sin 2 \theta } { \cos \theta } \right) ^ { 2 } = 6 \cot \theta - 4$$ giving your answers to 3 significant figures as appropriate.
3. Using the result from part (a), or otherwise, find the exact value of $$\int _ { \frac { \pi } { 6 } } ^ { \frac { \pi } { 4 } } \left( \frac { \cos 2 x } { \sin x } + \frac { \sin 2 x } { \cos x } \right) \cot x d x$$