Prove identity then solve

7

1. Show that \(\frac { \tan \theta } { 1 + \cos \theta } + \frac { \tan \theta } { 1 - \cos \theta } \equiv \frac { 2 } { \sin \theta \cos \theta }\).
2. Hence solve the equation \(\frac { \tan \theta } { 1 + \cos \theta } + \frac { \tan \theta } { 1 - \cos \theta } = \frac { 6 } { \tan \theta }\) for \(0 ^ { \circ } < \theta < 180 ^ { \circ }\).

4

1. Prove that \(\frac { ( \sin \theta + \cos \theta ) ^ { 2 } - 1 } { \cos ^ { 2 } \theta } \equiv 2 \tan \theta\). \includegraphics[max width=\textwidth, alt={}]{b5eb378d-a9cb-40e0-9203-374b58f1dcf9-05_63_1569_333_328} …........................................................................................................................................... \includegraphics[max width=\textwidth, alt={}, center]{b5eb378d-a9cb-40e0-9203-374b58f1dcf9-05_65_1570_511_324} \includegraphics[max width=\textwidth, alt={}, center]{b5eb378d-a9cb-40e0-9203-374b58f1dcf9-05_62_1570_603_324} \includegraphics[max width=\textwidth, alt={}, center]{b5eb378d-a9cb-40e0-9203-374b58f1dcf9-05_72_1570_685_324} \includegraphics[max width=\textwidth, alt={}]{b5eb378d-a9cb-40e0-9203-374b58f1dcf9-05_72_1570_776_324} ...................................................................................................................................... . \includegraphics[max width=\textwidth, alt={}]{b5eb378d-a9cb-40e0-9203-374b58f1dcf9-05_76_1572_952_322} ........................................................................................................................................ \includegraphics[max width=\textwidth, alt={}, center]{b5eb378d-a9cb-40e0-9203-374b58f1dcf9-05_72_1570_1137_324} \includegraphics[max width=\textwidth, alt={}, center]{b5eb378d-a9cb-40e0-9203-374b58f1dcf9-05_74_1572_1226_322} \includegraphics[max width=\textwidth, alt={}, center]{b5eb378d-a9cb-40e0-9203-374b58f1dcf9-05_77_1575_1315_319}
2. Hence solve the equation \(\frac { ( \sin \theta + \cos \theta ) ^ { 2 } - 1 } { \cos ^ { 2 } \theta } = 5 \tan ^ { 3 } \theta\) for \(- 90 ^ { \circ } < \theta < 90 ^ { \circ }\).

6

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

5

1. Show that \(\sin ^ { 4 } \theta - \cos ^ { 4 } \theta \equiv 2 \sin ^ { 2 } \theta - 1\).
2. Hence solve the equation \(\sin ^ { 4 } \theta - \cos ^ { 4 } \theta = \frac { 1 } { 2 }\) for \(0 ^ { \circ } \leqslant \theta \leqslant 360 ^ { \circ }\). \(6 \quad A\) is the point \(( a , 2 a - 1 )\) and \(B\) is the point \(( 2 a + 4,3 a + 9 )\), where \(a\) is a constant.
3. Find, in terms of \(a\), the gradient of a line perpendicular to \(A B\).
4. Given that the distance \(A B\) is \(\sqrt { } ( 260 )\), find the possible values of \(a\).

14. In this question solutions based entirely on graphical or numerical methods are not acceptable.

1. Solve, for \(- 180 ^ { \circ } \leqslant x < 180 ^ { \circ }\), the equation $$\sin \left( x + 60 ^ { \circ } \right) = - 0.4$$ giving your answers, in degrees, to one decimal place.
2. (a) Show that the equation $$2 \sin \theta \tan \theta - 3 = \cos \theta$$ can be written in the form $$3 \cos ^ { 2 } \theta + 3 \cos \theta - 2 = 0$$ (b) Hence solve, for \(0 \leqslant \theta < 360 ^ { \circ }\), the equation $$2 \sin \theta \tan \theta - 3 = \cos \theta$$ showing each stage of your working and giving your answers, in degrees, to one decimal place.

6. (a) Show that $$\frac { \cos ^ { 2 } x - \sin ^ { 2 } x } { 1 - \sin ^ { 2 } x } \equiv 1 - \tan ^ { 2 } x , \quad x \neq ( 2 n + 1 ) \frac { \pi } { 2 } , n \in \mathbb { Z }$$ (b) Hence solve, for \(0 \leqslant x < 2 \pi\), $$\frac { \cos ^ { 2 } x - \sin ^ { 2 } x } { 1 - \sin ^ { 2 } x } + 2 = 0$$ Give your answers in terms of \(\pi\).

12. (i) Solve, for \(0 < \theta \leqslant 360 ^ { \circ }\), $$3 \sin \left( \theta + 30 ^ { \circ } \right) = 2 \cos \left( \theta + 30 ^ { \circ } \right)$$ giving your answers, in degrees, to 2 decimal places.
(Solutions based entirely on graphical or numerical methods are not acceptable.)
(ii) (a) Given that $$\frac { \cos ^ { 2 } x + 2 \sin ^ { 2 } x } { 1 - \sin ^ { 2 } x } = 5$$ show that $$\tan ^ { 2 } x = k , \quad \text { where } k \text { is a constant. }$$ (b) Hence solve, for \(0 < x \leqslant 2 \pi\), $$\frac { \cos ^ { 2 } x + 2 \sin ^ { 2 } x } { 1 - \sin ^ { 2 } x } = 5$$ giving your answers, in radians, to 3 decimal places.

8. (i) Solve, for \(- 180 ^ { \circ } \leqslant x < 180 ^ { \circ }\), $$\tan \left( x - 40 ^ { \circ } \right) = 1.5$$ giving your answers to 1 decimal place.
(ii) (a) Show that the equation $$\sin \theta \tan \theta = 3 \cos \theta + 2$$ can be written in the form $$4 \cos ^ { 2 } \theta + 2 \cos \theta - 1 = 0$$ (b) Hence solve, for \(0 \leqslant \theta < 360 ^ { \circ }\), $$\sin \theta \tan \theta = 3 \cos \theta + 2$$ showing each stage of your working.

6

1. 1. Given that \(\tan 2 x + \tan x = 0\), show that \(\tan x = 0\) or \(\tan ^ { 2 } x = 3\).
   2. Hence find all solutions of \(\tan 2 x + \tan x = 0\) in the interval \(0 ^ { \circ } < x < 180 ^ { \circ }\).
      (l mark)
2. 1. Given that \(\cos x \neq 0\), show that the equation $$\sin 2 x = \cos x \cos 2 x$$ can be written in the form $$2 \sin ^ { 2 } x + 2 \sin x - 1 = 0$$
   2. Show that all solutions of the equation \(2 \sin ^ { 2 } x + 2 \sin x - 1 = 0\) are given by \(\sin x = \frac { \sqrt { 3 } - 1 } { p }\), where \(p\) is an integer.

3

1. Given that \(\sqrt { 2 \sin ^ { 2 } \theta + \cos \theta } = 2 \cos \theta\), show that \(6 \cos ^ { 2 } \theta - \cos \theta - 2 = 0\).
2. In this question you must show detailed reasoning. Solve the equation $$6 \cos ^ { 2 } \theta - \cos \theta - 2 = 0$$ giving all values of \(\theta\) between \(0 ^ { \circ }\) and \(360 ^ { \circ }\) correct to 1 decimal place.
3. Explain why not all the solutions from part (ii) are solutions of the equation $$\sqrt { 2 \sin ^ { 2 } \theta + \cos \theta } = 2 \cos \theta$$

8. (a) Prove that $$\frac { 1 - \cos 2 \theta } { \sin 2 \theta } \equiv \tan \theta , \theta \neq \frac { n \pi } { 2 } , \quad n \in \mathbb { Z }$$ (b) Solve, giving exact answers in terms of \(\pi\), $$2 ( 1 - \cos 2 \theta ) = \tan \theta , \quad 0 < \theta < \pi$$ [P2 January 2002 Question 6]