1.05p Proof involving trig: functions and identities

5

1. Prove the identity \(\cos 4 \theta - 4 \cos 2 \theta \equiv 8 \sin ^ { 4 } \theta - 3\).
2. Hence solve the equation $$\cos 4 \theta = 4 \cos 2 \theta + 3$$ for \(0 ^ { \circ } \leqslant \theta \leqslant 360 ^ { \circ }\).

5

1. Prove the identity \(\cos 4 \theta - 4 \cos 2 \theta + 3 \equiv 8 \sin ^ { 4 } \theta\).
2. Using this result find, in simplified form, the exact value of $$\int _ { \frac { 1 } { 6 } \pi } ^ { \frac { 1 } { 3 } \pi } \sin ^ { 4 } \theta \mathrm {~d} \theta$$

5

1. By differentiating \(\frac { 1 } { \cos x }\), show that if \(y = \sec x\) then \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \sec x \tan x\).
2. Show that \(\frac { 1 } { \sec x - \tan x } \equiv \sec x + \tan x\).
3. Deduce that \(\frac { 1 } { ( \sec x - \tan x ) ^ { 2 } } \equiv 2 \sec ^ { 2 } x - 1 + 2 \sec x \tan x\).
4. Hence show that \(\int _ { 0 } ^ { \frac { 1 } { 4 } \pi } \frac { 1 } { ( \sec x - \tan x ) ^ { 2 } } \mathrm {~d} x = \frac { 1 } { 4 } ( 8 \sqrt { } 2 - \pi )\).

5

1. Prove that \(\cot \theta + \tan \theta \equiv 2 \operatorname { cosec } 2 \theta\).
2. Hence show that \(\int _ { \frac { 1 } { 6 } \pi } ^ { \frac { 1 } { 3 } \pi } \operatorname { cosec } 2 \theta \mathrm {~d} \theta = \frac { 1 } { 2 } \ln 3\).

5

1. Prove the identity \(\tan 2 \theta - \tan \theta \equiv \tan \theta \sec 2 \theta\).
2. Hence show that \(\int _ { 0 } ^ { \frac { 1 } { 6 } \pi } \tan \theta \sec 2 \theta \mathrm {~d} \theta = \frac { 1 } { 2 } \ln \frac { 3 } { 2 }\).

4

1. Prove the identity $$\tan \left( x + 45 ^ { \circ } \right) - \tan \left( 45 ^ { \circ } - x \right) \equiv 2 \tan 2 x .$$
2. Hence solve the equation $$\tan \left( x + 45 ^ { \circ } \right) - \tan \left( 45 ^ { \circ } - x \right) = 2 ,$$ for \(0 ^ { \circ } \leqslant x \leqslant 180 ^ { \circ }\).

8

1. (a) Prove the identity $$\sec ^ { 2 } x + \sec x \tan x \equiv \frac { 1 + \sin x } { \cos ^ { 2 } x }$$ (b) Hence prove that $$\sec ^ { 2 } x + \sec x \tan x \equiv \frac { 1 } { 1 - \sin x }$$
2. By differentiating \(\frac { 1 } { \cos x }\), show that if \(y = \sec x\) then \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \sec x \tan x\).
3. Using the results of parts (i) and (ii), find the exact value of $$\int _ { 0 } ^ { \frac { 1 } { 4 } \pi } \frac { 1 } { 1 - \sin x } \mathrm {~d} x$$

6

1. Prove that \(\operatorname { cosec } 2 \theta - \cot 2 \theta \equiv \tan \theta\).
2. Hence show that \(\int _ { \frac { 1 } { 4 } \pi } ^ { \frac { 1 } { 3 } \pi } ( \operatorname { cosec } 2 \theta - \cot 2 \theta ) \mathrm { d } \theta = \frac { 1 } { 2 } \ln 2\).

9. In this question you must show detailed reasoning. Solutions relying entirely on calculator technology are not acceptable.

1. Solve, for \(0 < x \leqslant \pi\), the equation $$2 \sec ^ { 2 } x - 3 \tan x = 2$$ giving the answers, as appropriate, to 3 significant figures.
2. Prove that $$\frac { \sin 3 \theta } { \sin \theta } - \frac { \cos 3 \theta } { \cos \theta } \equiv 2$$

   VIAV SIHI NI III IM I ON OC

   VIIIV SIHI NI JIIIM I ON OO

   VAYV SIHI NI JIIYM ION OC

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 the equation $$\frac { 3 \sin \theta \cos \theta } { \cos \theta + \sin \theta } = ( 2 + \sec 2 \theta ) ( \cos \theta - \sin \theta )$$ can be written in the form $$3 \sin 2 \theta - 4 \cos 2 \theta = 2$$
2. Hence solve for \(\pi < x < \frac { 3 \pi } { 2 }\) $$\frac { 3 \sin x \cos x } { \cos x + \sin x } = ( 2 + \sec 2 x ) ( \cos x - \sin x )$$ giving the answer to 3 significant figures.

8. (a) Using the trigonometric identity for \(\tan ( A + B )\), prove that $$\tan 3 x = \frac { 3 \tan x - \tan ^ { 3 } x } { 1 - 3 \tan ^ { 2 } x } , \quad x \neq ( 2 n + 1 ) 30 ^ { \circ } , \quad n \in \mathbb { Z }$$ (b) Hence solve, for \(- 30 ^ { \circ } < x < 30 ^ { \circ }\), $$\tan 3 x = 11 \tan x$$ (Solutions based entirely on graphical or numerical methods are not acceptable.)

4. (a) Prove that $$\frac { 1 - \cos 2 x } { \sin 2 x } \equiv \tan x , \quad x \neq \frac { n \pi } { 2 }$$ (b) Hence solve, for \(0 \leqslant \theta < 2 \pi\), $$3 \sec ^ { 2 } \theta - 7 = \frac { 1 - \cos 2 \theta } { \sin 2 \theta }$$ Give your answers in radians to 3 decimal places, as appropriate.
(Solutions based entirely on graphical or numerical methods are not acceptable.)

7. (i) (a) Prove that $$\cos 3 \theta \equiv 4 \cos ^ { 3 } \theta - 3 \cos \theta$$ (You may use the double angle formulae and the identity $$\cos ( A + B ) \equiv \cos A \cos B - \sin A \sin B )$$ (b) Hence solve the equation $$2 \cos 3 \theta + \cos 2 \theta + 1 = 0$$ giving answers in the interval \(0 \leqslant \theta \leqslant \pi\).
Solutions based entirely on graphical or numerical methods are not acceptable.
(ii) Given that \(\theta = \arcsin x\) and that \(0 < \theta < \frac { \pi } { 2 }\), show that $$\cot \theta = \frac { \sqrt { \left( 1 - x ^ { 2 } \right) } } { x } , \quad 0 < x < 1$$

1. (a) Using \(\sin ^ { 2 } \theta + \cos ^ { 2 } \theta \equiv 1\), show that \(\operatorname { cosec } ^ { 2 } \theta - \cot ^ { 2 } \theta \equiv 1\).
   (b) Hence, or otherwise, prove that

$$\operatorname { cosec } ^ { 4 } \theta - \cot ^ { 4 } \theta \equiv \operatorname { cosec } ^ { 2 } \theta + \cot ^ { 2 } \theta$$ (c) Solve, for \(90 ^ { \circ } < \theta < 180 ^ { \circ }\), $$\operatorname { cosec } ^ { 4 } \theta - \cot ^ { 4 } \theta = 2 - \cot \theta$$

6. (a) Prove that $$\frac { 1 } { \sin 2 \theta } - \frac { \cos 2 \theta } { \sin 2 \theta } = \tan \theta , \quad \theta \neq 90 n ^ { \circ } , n \in \mathbb { Z }$$ (b) Hence, or otherwise,

1. show that \(\tan 15 ^ { \circ } = 2 - \sqrt { 3 }\),
2. solve, for \(0 < x < 360 ^ { \circ }\), $$\operatorname { cosec } 4 x - \cot 4 x = 1$$

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 \(\pi\).

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

$$\sec 2 A + \tan 2 A \equiv \frac { \cos A + \sin A } { \cos A - \sin A } , \quad A \neq \frac { ( 2 n + 1 ) \pi } { 4 } , n \in \mathbb { Z }$$ (b) Hence solve, for \(0 \leqslant \theta < 2 \pi\), $$\sec 2 \theta + \tan 2 \theta = \frac { 1 } { 2 }$$ Give your answers to 3 decimal places.

3. (a) Using the identity for \(\cos ( A + B )\), prove that \(\cos \theta \equiv 1 - 2 \sin ^ { 2 } \left( \frac { 1 } { 2 } \theta \right)\).
(b) Prove that \(1 + \sin \theta - \cos \theta \equiv 2 \sin \left( \frac { 1 } { 2 } \theta \right) \left[ \cos \left( \frac { 1 } { 2 } \theta \right) + \sin \left( \frac { 1 } { 2 } \theta \right) \right]\).
(c) Hence, or otherwise, solve the equation $$1 + \sin \theta - \cos \theta = 0 , \quad 0 \leq \theta < 2 \pi$$

7. (i) Prove that, for \(\cos x \neq 0\), $$\sin 2 x - \tan x \equiv \tan x \cos 2 x$$ (ii) 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. (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

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 \(\frac { 4 \cos 2 x } { 1 + \cos 2 x } = 3 \tan x - 7\).

9

1. Prove the identity $$\tan \left( \theta + 60 ^ { \circ } \right) \tan \left( \theta - 60 ^ { \circ } \right) \equiv \frac { \tan ^ { 2 } \theta - 3 } { 1 - 3 \tan ^ { 2 } \theta }$$
2. Solve, for \(0 ^ { \circ } < \theta < 180 ^ { \circ }\), the equation $$\tan \left( \theta + 60 ^ { \circ } \right) \tan \left( \theta - 60 ^ { \circ } \right) = 4 \sec ^ { 2 } \theta - 3 ,$$ giving your answers correct to the nearest \(0.1 ^ { \circ }\).
3. Show that, for all values of the constant k , the equation $$\tan \left( \theta + 60 ^ { \circ } \right) \tan \left( \theta - 60 ^ { \circ } \right) = \mathrm { K } ^ { 2 }$$ has two roots in the interval \(0 ^ { \circ } < \theta < 180 ^ { \circ }\).

6 Prove that

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