Prove identity then solve equation

7. (a) Show that $$\operatorname { cosec } 2 x + \cot 2 x = \cot x , \quad x \neq 90 n ^ { \circ } , \quad n \in \mathbb { Z }$$ (b) Hence, or otherwise, solve, for \(0 \leqslant \theta < 180 ^ { \circ }\), $$\operatorname { cosec } \left( 4 \theta + 10 ^ { \circ } \right) + \cot \left( 4 \theta + 10 ^ { \circ } \right) = \sqrt { 3 }$$ You must show your working.
(Solutions based entirely on graphical or numerical methods are not acceptable.)

1. (a) Prove that

$$2 \cot 2 x + \tan x \equiv \cot x \quad x \neq \frac { n \pi } { 2 } , n \in \mathbb { Z }$$ (b) Hence, or otherwise, solve, for \(- \pi \leqslant x < \pi\), $$6 \cot 2 x + 3 \tan x = \operatorname { cosec } ^ { 2 } x - 2$$ Give your answers to 3 decimal places.
(Solutions based entirely on graphical or numerical methods are not acceptable.) \includegraphics[max width=\textwidth, alt={}, center]{d3ba2776-eedb-48f0-834f-41aa454afba3-14_2258_47_315_37}

6. (i) Prove the identity $$2 \cot 2 x + \tan x \equiv \cot x , \quad x \neq \frac { n } { 2 } \pi , \quad n \in \mathbb { Z }$$ (ii) Solve, for \(0 \leq x < \pi\), the equation $$2 \cot 2 x + \tan x = \operatorname { cosec } ^ { 2 } x - 7$$ giving your answers to 2 decimal places.

7. (i) Use the identity $$\cos ( A + B ) \equiv \cos A \cos B - \sin A \sin B$$ to prove that $$\cos x \equiv 1 - 2 \sin ^ { 2 } \frac { x } { 2 }$$ (ii) Prove that, for \(\sin x \neq 0\), $$\frac { 1 - \cos x } { \sin x } \equiv \tan \frac { x } { 2 }$$ (iii) Find the values of \(x\) in the interval \(0 \leq x \leq 360 ^ { \circ }\) for which $$\frac { 1 - \cos x } { \sin x } = 2 \sec ^ { 2 } \frac { x } { 2 } - 5$$ giving your answers to 1 decimal place where appropriate.

7

1. Write down the formula for \(\cos 2 x\) in terms of \(\cos x\).
2. Prove the identity \(\frac { 4 \cos 2 x } { 1 + \cos 2 x } \equiv 4 - 2 \sec ^ { 2 } x\).
3. Solve, for \(0 < x < 2 \pi\), the equation \(\frac { 4 \cos 2 x } { 1 + \cos 2 x } = 3 \tan x - 7\).

4 Show that \(\frac { 1 + \tan ^ { 2 } \theta } { 1 - \tan ^ { 2 } \theta } = \sec 2 \theta\).
Hence, or otherwise, solve the equation \(\frac { 1 + \tan ^ { 2 } \theta } { 1 - \tan ^ { 2 } \theta } = 2\), for \(0 ^ { \circ } \leqslant \theta \leqslant 180 ^ { \circ }\).

2

1. Show that \(\cos ( \alpha + \beta ) = \frac { 1 - \tan \alpha \tan \beta } { \sec \alpha \sec \beta }\).
2. Hence show that \(\cos 2 \alpha = \frac { 1 - \tan ^ { 2 } \alpha } { 1 + \tan ^ { 2 } \alpha }\).
3. Hence or otherwise solve the equation \(\frac { 1 - \tan ^ { 2 } \theta } { 1 + \tan ^ { 2 } \theta } = \frac { 1 } { 2 }\) for \(0 ^ { \circ } \leqslant \theta \leqslant 180 ^ { \circ }\).

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

8. (a) Prove that $$\sin 2 x - \tan x \equiv \tan x \cos 2 x , \quad x \neq \frac { ( 2 n + 1 ) \pi } { 2 } , \quad n \in \mathbb { Z }$$ (b) Hence solve, for \(0 \leqslant \theta < \frac { \pi } { 2 }\)

1. \(\sin 2 \theta - \tan \theta = \sqrt { 3 } \cos 2 \theta\)
2. \(\tan ( \theta + 1 ) \cos ( 2 \theta + 2 ) - \sin ( 2 \theta + 2 ) = 2\) Give your answers in radians to 3 significant figures, as appropriate.
   (Solutions based entirely on graphical or numerical methods are not acceptable.)

7

1. Show that \(\frac { \sin ^ { 2 } x - \cos ^ { 2 } x } { 1 - \sin ^ { 2 } x } \equiv \tan ^ { 2 } x - 1\).
2. Hence solve the equation $$\frac { \sin ^ { 2 } x - \cos ^ { 2 } x } { 1 - \sin ^ { 2 } x } = 5 - \tan x$$ for \(0 ^ { \circ } \leqslant x \leqslant 360 ^ { \circ }\).

8 The functions f and g are defined for all real values of \(x\) by $$\mathrm { f } ( x ) = | 2 x + a | + 3 a \quad \text { and } \quad \mathrm { g } ( x ) = 5 x - 4 a$$ where \(a\) is a positive constant.

1. State the range of f and the range of g .
2. State why f has no inverse, and find an expression for \(\mathrm { g } ^ { - 1 } ( x )\).
3. Solve for \(x\) the equation \(\operatorname { gf } ( x ) = 31 a\).
4. Show that \(\sin 2 \theta ( \tan \theta + \cot \theta ) \equiv 2\).
5. Hence
   1. find the exact value of \(\tan \frac { 1 } { 12 } \pi + \tan \frac { 1 } { 8 } \pi + \cot \frac { 1 } { 12 } \pi + \cot \frac { 1 } { 8 } \pi\),
   2. solve the equation \(\sin 4 \theta ( \tan \theta + \cot \theta ) = 1\) for \(0 < \theta < \frac { 1 } { 2 } \pi\),
   3. express \(( 1 - \cos 2 \theta ) ^ { 2 } \left( \tan \frac { 1 } { 2 } \theta + \cot \frac { 1 } { 2 } \theta \right) ^ { 3 }\) in terms of \(\sin \theta\).

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

Solutions relying entirely on calculator technology are not acceptable.

1. Show that $$\frac { 1 } { \cos \theta } + \tan \theta \equiv \frac { \cos \theta } { 1 - \sin \theta } \quad \theta \neq ( 2 n + 1 ) 90 ^ { \circ } \quad n \in \mathbb { Z }$$ Given that \(\cos 2 x \neq 0\)
2. solve for \(0 < x < 90 ^ { \circ }\) $$\frac { 1 } { \cos 2 x } + \tan 2 x = 3 \cos 2 x$$ giving your answers to one decimal place.

12. a. Show that $$\sec \theta - \cos \theta = \sin \theta \tan \theta \quad \theta \neq ( \pi n ) ^ { 0 } \quad n \in Z$$ b. Hence, or otherwise, solve for \(0 < x \leq \pi\) $$\sec x - \cos x = \sin x \tan \left( 3 x - \frac { \pi } { 9 } \right)$$

12. a. Prove that $$\frac { \sec ^ { 2 } x - 1 } { \sec ^ { 2 } x } \equiv \sin ^ { 2 } x$$ b. Hence solve, for \(- 360 ^ { \circ } < x < 360 ^ { \circ }\), the equation $$\frac { \sec ^ { 2 } x - 1 } { \sec ^ { 2 } x } = \frac { \cos 2 x } { 2 }$$

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

Solutions relying entirely on calculator technology are not acceptable.

1. Show that $$\operatorname { cosec } \theta - \sin \theta \equiv \cos \theta \cot \theta \quad \theta \neq ( 180 n ) ^ { \circ } \quad n \in \mathbb { Z }$$
2. Hence, or otherwise, solve for \(0 < x < 180 ^ { \circ }\) $$\operatorname { cosec } x - \sin x = \cos x \cot \left( 3 x - 50 ^ { \circ } \right)$$

1. (a) Prove

$$\frac { \cos 3 \theta } { \sin \theta } + \frac { \sin 3 \theta } { \cos \theta } \equiv 2 \cot 2 \theta \quad \theta \neq ( 90 n ) ^ { \circ } , n \in \mathbb { Z }$$ (b) Hence solve, for \(90 ^ { \circ } < \theta < 180 ^ { \circ }\), the equation $$\frac { \cos 3 \theta } { \sin \theta } + \frac { \sin 3 \theta } { \cos \theta } = 4$$ giving any solutions to one decimal place.

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

Solutions relying entirely on calculator technology are not acceptable.

1. Prove that $$\frac { 1 } { \operatorname { cosec } \theta - 1 } + \frac { 1 } { \operatorname { cosec } \theta + 1 } \equiv 2 \tan \theta \sec \theta \quad \theta \neq ( 90 n ) ^ { \circ } , n \in \mathbb { Z }$$
2. Hence solve, for \(0 < x < 90 ^ { \circ }\), the equation $$\frac { 1 } { \operatorname { cosec } 2 x - 1 } + \frac { 1 } { \operatorname { cosec } 2 x + 1 } = \cot 2 x \sec 2 x$$ Give each answer, in degrees, to one decimal place.

13. (a) Show that $$\operatorname { cosec } 2 x + \cot 2 x \equiv \cot x , \quad x \neq 90 n ^ { \circ } , n \in \mathbb { Z }$$ (b) Hence, or otherwise, solve, for \(0 \leqslant \theta < 180 ^ { \circ }\), $$\operatorname { cosec } \left( 4 \theta + 10 ^ { \circ } \right) + \cot \left( 4 \theta + 10 ^ { \circ } \right) = \sqrt { 3 }$$ You must show your working.
(Solutions based entirely on graphical or numerical methods are not acceptable.)

6

1. Prove that \(\frac { \sin \theta } { 1 - \cos \theta } - \frac { 1 } { \sin \theta } = \cot \theta\).
2. Hence find the exact roots of the equation \(\frac { \sin \theta } { 1 - \cos \theta } - \frac { 1 } { \sin \theta } = 3 \tan \theta\) in the interval \(0 \leqslant \theta \leqslant \pi\). Answer all the questions.
   Section B (75 marks)

8

1. Show that the expression \(\frac { 1 - \sin x } { \cos x } + \frac { \cos x } { 1 - \sin x }\) can be written as \(2 \sec x\).
   [0pt] [4 marks]
2. Hence solve the equation $$\frac { 1 - \sin x } { \cos x } + \frac { \cos x } { 1 - \sin x } = \tan ^ { 2 } x - 2$$ giving the values of \(x\) to the nearest degree in the interval \(0 ^ { \circ } \leqslant x < 360 ^ { \circ }\).
   [0pt] [6 marks]
3. Hence solve the equation $$\frac { 1 - \sin \left( 2 \theta - 30 ^ { \circ } \right) } { \cos \left( 2 \theta - 30 ^ { \circ } \right) } + \frac { \cos \left( 2 \theta - 30 ^ { \circ } \right) } { 1 - \sin \left( 2 \theta - 30 ^ { \circ } \right) } = \tan ^ { 2 } \left( 2 \theta - 30 ^ { \circ } \right) - 2$$ giving the values of \(\theta\) to the nearest degree in the interval \(0 ^ { \circ } \leqslant \theta \leqslant 180 ^ { \circ }\).
   [0pt] [2 marks]
   \includegraphics[max width=\textwidth, alt={}]{57412ec0-ad97-4418-8ba8-93f1f7d8aac1-16_1517_1709_1190_153}
   \includegraphics[max width=\textwidth, alt={}]{57412ec0-ad97-4418-8ba8-93f1f7d8aac1-20_2489_1730_221_139}

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

6. (a) Prove the identity $$2 \cot 2 x + \tan x \equiv \cot x , \quad x \neq \frac { n } { 2 } \pi , \quad n \in \mathbb { Z } .$$ (b) Solve, for \(0 \leq x < \pi\), the equation $$2 \cot 2 x + \tan x = \operatorname { cosec } ^ { 2 } x - 7 ,$$ giving your answers to 2 decimal places.