Prove algebraic trigonometric identity

3 Prove the identity \(\frac { 1 - \tan ^ { 2 } x } { 1 + \tan ^ { 2 } x } \equiv 1 - 2 \sin ^ { 2 } x\).

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

2 Prove the identity $$\frac { 1 + \sin x } { \cos x } + \frac { \cos x } { 1 + \sin x } \equiv \frac { 2 } { \cos x }$$

2 Prove the identity $$\tan ^ { 2 } x - \sin ^ { 2 } x \equiv \tan ^ { 2 } x \sin ^ { 2 } x$$

7. (i) Show that $$\tan \theta + \frac { 1 } { \tan \theta } \equiv \frac { 1 } { \sin \theta \cos \theta } \quad \theta \neq \frac { \mathrm { n } \pi } { 2 } \quad n \in \mathbb { Z }$$ (ii) Solve, for \(0 \leqslant x < 90 ^ { \circ }\), the equation $$3 \cos ^ { 2 } \left( 2 x + 10 ^ { \circ } \right) = 1$$ giving your answers in degrees to one decimal place.
(Solutions based entirely on graphical or numerical methods are not acceptable.)

1. 1. Prove that

$$\sec ^ { 2 } x - \operatorname { cosec } ^ { 2 } x \equiv \tan ^ { 2 } x - \cot ^ { 2 } x$$ (ii) Given that $$y = \arccos x , \quad - 1 \leqslant x \leqslant 1 \text { and } 0 \leqslant y \leqslant \pi ,$$

1. express arcsin \(x\) in terms of \(y\).
2. Hence evaluate \(\arccos x + \arcsin x\). Give your answer in terms of \(\pi\).

6 Prove that

1. \(\frac { \sin 2 \theta } { 2 \tan \theta } + \sin ^ { 2 } \theta = 1\),
2. \(\quad \sin \left( x + 45 ^ { \circ } \right) = \cos \left( x - 45 ^ { \circ } \right)\).

4 Prove that \(\sec ^ { 2 } \theta + \operatorname { cosec } ^ { 2 } \theta = \sec ^ { 2 } \theta \operatorname { cosec } ^ { 2 } \theta\).

5 Prove that \(\cot \beta - \cot \alpha = \frac { \sin ( \alpha - \beta ) } { \sin \alpha \sin \beta }\).

9

1. By first expanding \(\cos ( 2 \theta + \theta )\), prove that $$\cos 3 \theta \equiv 4 \cos ^ { 3 } \theta - 3 \cos \theta$$
2. Hence prove that $$\cos 6 \theta \equiv 32 \cos ^ { 6 } \theta - 48 \cos ^ { 4 } \theta + 18 \cos ^ { 2 } \theta - 1$$
3. Show that the only solutions of the equation $$1 + \cos 6 \theta = 18 \cos ^ { 2 } \theta$$ are odd multiples of \(90 ^ { \circ }\).

4 Prove that \(\cot \beta - \cot \alpha = \frac { \sin ( \alpha - \beta ) } { \sin \alpha \sin \beta }\).

1. (a) Prove that

$$\tan \theta + \cot \theta \equiv 2 \operatorname { cosec } 2 \theta , \quad \theta \neq \frac { n \pi } { 2 } , n \in \mathbb { Z }$$ (b) Hence explain why the equation $$\tan \theta + \cot \theta = 1$$ does not have any real solutions.

10 The diagram shows the graph of \(\mathrm { y } = 1.5 + \sin ^ { 2 } \mathrm { x }\) for \(0 \leqslant x \leqslant 2 \pi\). \includegraphics[max width=\textwidth, alt={}, center]{8eeff88d-8b05-43c6-86a5-bd82221c0bea-07_512_1278_322_242}

1. Show that the equation of the graph can be written in the form \(\mathrm { y } = \mathrm { a } - \mathrm { b } \cos 2 \mathrm { x }\) where \(a\) and \(b\) are constants to be determined.
2. Write down the period of the function \(1.5 + \sin ^ { 2 } x\).
3. Determine the \(x\)-coordinates of the points of intersection of the graph of \(y = 1.5 + \sin ^ { 2 } x\) with the graph of \(\mathrm { y } = 1 + \cos 2 \mathrm { x }\) in the interval \(0 \leqslant x \leqslant 2 \pi\).

10

1. You are given that \(\left( x ^ { 2 } + y ^ { 2 } \right) ^ { 3 } = x ^ { 6 } + 3 x ^ { 4 } y ^ { 2 } + 3 x ^ { 2 } y ^ { 4 } + y ^ { 6 }\).
   Hence, or otherwise, prove that \(\sin ^ { 6 } \theta + \cos ^ { 6 } \theta = 1 - \frac { 3 } { 4 } \sin ^ { 2 } 2 \theta\) for all values of \(\theta\).
2. Use the result from part (a) to determine the minimum value of \(\sin ^ { 6 } \theta + \cos ^ { 6 } \theta\). The questions in this section refer to the article on the Insert. You should read the article before attempting the questions.

7 Prove that \(\sin 8 \theta \tan 4 \theta + \cos 8 \theta = 1\).

8 Prove that, for all values of \(x\), the value of the expression $$( 3 \sin x + \cos x ) ^ { 2 } + ( \sin x - 3 \cos x ) ^ { 2 }$$ is an integer and state its value.

8

1. Given that \(\cos \theta \neq \pm 1\), prove the identity $$\frac { 1 } { 1 - \cos \theta } + \frac { 1 } { 1 + \cos \theta } \equiv 2 \operatorname { cosec } ^ { 2 } \theta$$ 8
2. Hence, find the set of values of \(A\) for which the equation $$\frac { 1 } { 1 - \cos \theta } + \frac { 1 } { 1 + \cos \theta } = A$$ has real solutions.
   Fully justify your answer.
   8
3. Given that \(\theta\) is obtuse and $$\frac { 1 } { 1 - \cos \theta } + \frac { 1 } { 1 + \cos \theta } = 16$$ find the exact value of \(\cot \theta\)

Prove that \(\sec^2\theta + \cosec^2\theta = \sec^2\theta \cosec^2\theta\). [4]

One of the equations below is true for all values of \(x\) Identify the correct equation. Tick (\(\checkmark\)) one box. [1 mark] \(\cos^2 x = -1 - \sin^2 x\) \(\square\) \(\cos^2 x = -1 + \sin^2 x\) \(\square\) \(\cos^2 x = 1 - \sin^2 x\) \(\square\) \(\cos^2 x = 1 + \sin^2 x\) \(\square\)