Prove identity with double/compound angles

1. (a) Prove that

$$\sin 2 x - \tan x \equiv \tan x \cos 2 x , \quad x \neq ( 2 n + 1 ) 90 ^ { \circ } , \quad n \in \mathbb { Z }$$ (b) Given that \(x \neq 90 ^ { \circ }\) and \(x \neq 270 ^ { \circ }\), solve, for \(0 \leqslant x < 360 ^ { \circ }\), $$\sin 2 x - \tan x = 3 \tan x \sin x$$ Give your answers in degrees to one decimal place where appropriate.
(Solutions based entirely on graphical or numerical methods are not acceptable.)

5 Show that \(\frac { \sin 2 \theta } { 1 + \cos 2 \theta } = \tan \theta\).

10

1. \includegraphics[max width=\textwidth, alt={}, center]{6b766f5c-8533-4e0c-bb10-0d9949dc777b-7_599_780_267_328} The diagram shows triangle \(A B C\). The perpendicular from \(C\) to \(A B\) meets \(A B\) at \(D\). Angle \(A C D = x\), angle \(D C B = y\), length \(B C = a\) and length \(A C = b\).
   1. Explain why the length of \(C D\) can be written as \(a \cos y\).
   2. Show that the area of the triangle \(A D C\) is given by \(\frac { 1 } { 2 } a b \sin x \cos y\).
   3. Hence, or otherwise, show that \(\sin ( x + y ) = \sin x \cos y + \cos x \sin y\).
2. Given that \(\sin \left( 30 ^ { \circ } + \alpha \right) = \cos \left( 45 ^ { \circ } - \alpha \right)\), show that \(\tan \alpha = 2 + \sqrt { 6 } - \sqrt { 3 } - \sqrt { 2 }\).

7

1. Use the result \(\cos ( A + B ) = \cos A \cos B - \sin A \sin B\) to show that \(\cos ( A - B ) = \cos A \cos B + \sin A \sin B\). The function \(\mathrm { f } ( \theta )\) is defined as \(\cos \left( \theta + 30 ^ { \circ } \right) \cos \left( \theta - 30 ^ { \circ } \right)\), where \(\theta\) is in degrees.
2. Show that \(f ( \theta ) = \cos ^ { 2 } \theta - \frac { 1 } { 4 }\).
3. (i) Determine the following.

4 Prove that \(\sin ^ { 2 } ( \theta + 45 ) ^ { \circ } - \cos ^ { 2 } ( \theta + 45 ) ^ { \circ } \equiv \sin 2 \theta ^ { \circ }\).

7

1. Use the identity $$\tan ( A + B ) = \frac { \tan A + \tan B } { 1 - \tan A \tan B }$$ to express \(\tan 2 x\) in terms of \(\tan x\).
2. Show that $$2 - 2 \tan x - \frac { 2 \tan x } { \tan 2 x } = ( 1 - \tan x ) ^ { 2 }$$ for all values of \(x , \tan 2 x \neq 0\).

4

1. 1. Express \(\sin 2 x\) in terms of \(\sin x\) and \(\cos x\).
   2. Express \(\cos 2 x\) in terms of \(\cos x\).
2. Show that $$\sin 2 x - \tan x = \tan x \cos 2 x$$ for all values of \(x\).
3. Solve the equation \(\sin 2 x - \tan x = 0\), giving all solutions in degrees in the interval \(0 ^ { \circ } < x < 360 ^ { \circ }\).

7

1. Use the identity \(\tan ( A - B ) = \frac { \tan A - \tan B } { 1 + \tan A \tan B }\) with \(A = ( r + 1 ) x\) and \(B = r x\) to show that $$\tan r x \tan ( r + 1 ) x = \frac { \tan ( r + 1 ) x } { \tan x } - \frac { \tan r x } { \tan x } - 1$$ (4 marks)
2. Use the method of differences to show that $$\tan \frac { \pi } { 50 } \tan \frac { 2 \pi } { 50 } + \tan \frac { 2 \pi } { 50 } \tan \frac { 3 \pi } { 50 } + \ldots + \tan \frac { 19 \pi } { 50 } \tan \frac { 20 \pi } { 50 } = \frac { \tan \frac { 2 \pi } { 5 } } { \tan \frac { \pi } { 50 } } - 20$$

14 Some students are trying to prove an identity for \(\sin ( A + B )\). They start by drawing two right-angled triangles \(O D E\) and \(O E F\), as shown. \includegraphics[max width=\textwidth, alt={}, center]{85b10472-8149-4387-999f-2ef153f1a105-22_695_662_477_689} The students' incomplete proof continues,
Let angle \(D O E = A\) and angle \(E O F = B\).
In triangle OFR,
Line \(1 \quad \sin ( A + B ) = \frac { R F } { O F }\) Line 2 $$= \frac { R P + P F } { O F }$$ Line 3 $$= \frac { D E } { O F } + \frac { P F } { O F } \text { since } D E = R P$$ Line 4 $$= \frac { D E } { \cdots \cdots } \times \frac { \cdots \cdots } { O F } + \frac { P F } { E F } \times \frac { E F } { O F }$$ Line 5 \(=\) \(\_\_\_\_\) \(+ \cos A \sin B\) 14

1. Explain why \(\frac { P F } { E F } \times \frac { E F } { O F }\) in Line 4 leads to \(\cos A \sin B\) in Line 5
   14
2. Complete Line 4 and Line 5 to prove the identity Line 4 $$= \frac { D E } { \ldots \ldots } \times \frac { \cdots \ldots } { O F } + \frac { P F } { E F } \times \frac { E F } { O F }$$ Line 5 = \(+ \cos A \sin B\) 14
3. Explain why the argument used in part (a) only proves the identity when \(A\) and \(B\) are acute angles. 14
4. Another student claims that by replacing \(B\) with \(- B\) in the identity for \(\sin ( A + B )\) it is possible to find an identity for \(\sin ( A - B )\). Assuming the identity for \(\sin ( A + B )\) is correct for all values of \(A\) and \(B\), prove a similar result for \(\sin ( A - B )\).

9 A robotic arm which is attached to a flat surface at the origin \(O\), is used to draw a graphic design. The arm is made from two rods \(O P\) and \(P Q\), each of length \(d\), which are joined at \(P\).
A pen is attached to the arm at \(Q\).
The coordinates of the pen are controlled by adjusting the angle \(O P Q\) and the angle \(\theta\) between \(O P\) and the \(x\)-axis. For this particular design the pen is made to move so that the two angles are always equal to each other with \(0 \leq \theta \leq \frac { \pi } { 2 }\) as shown in Figure 2. \begin{figure}[h] \end{figure} 9

1. Show that the \(x\)-coordinate of the pen can be modelled by the equation $$x = d \left( \cos \theta + \sin \left( 2 \theta - \frac { \pi } { 2 } \right) \right)$$ 9
2. Hence, show that $$x = d \left( 1 + \cos \theta - 2 \cos ^ { 2 } \theta \right)$$ 9
3. It can be shown that $$x = \frac { 9 d } { 8 } - d \left( \cos \theta - \frac { 1 } { 4 } \right) ^ { 2 }$$ State the greatest possible value of \(x\) and the corresponding value of \(\cos \theta\) 9
4. Figure 3 below shows the arm when the \(x\)-coordinate is at its greatest possible value. \begin{figure}[h]
   \captionsetup{labelformat=empty} \caption{Figure 3} \includegraphics[alt={},max width=\textwidth]{b7df05bf-f3fc-4705-a13c-6b562896fa9f-14_570_773_456_630}
   \end{figure} Find, in terms of \(d\), the exact distance \(O Q\). \includegraphics[max width=\textwidth, alt={}, center]{b7df05bf-f3fc-4705-a13c-6b562896fa9f-15_2488_1716_219_153}

3 By expanding \(\left( z ^ { 2 } + \frac { 1 } { z ^ { 2 } } \right) ^ { 3 }\), where \(z = \mathrm { e } ^ { \mathrm { i } \theta }\), show that \(4 \cos ^ { 3 } 2 \theta = \cos 6 \theta + 3 \cos 2 \theta\).

12 For each positive integer \(n\), the function \(\mathrm { F } _ { n }\) is defined for all real angles \(\theta\) by $$\mathrm { F } _ { n } ( \theta ) = c ^ { 2 n } + s ^ { 2 n }$$ where \(c = \cos \theta\) and \(s = \sin \theta\).

1. Prove the identity $$\mathrm { F } _ { n + 2 } ( \theta ) - \frac { 1 } { 4 } \sin ^ { 2 } 2 \theta \times \mathrm { F } _ { n + 1 } ( \theta ) \equiv \mathrm { F } _ { n + 3 } ( \theta )$$ Let \(z\) denote the complex number \(c + \mathrm { i } s\).
2. Using de Moivre's theorem,
   1. express \(z + z ^ { - 1 }\) and \(z - z ^ { - 1 }\) in terms of \(c\) and \(s\) respectively,
   2. prove the identity \(8 \left( c ^ { 6 } + s ^ { 6 } \right) \equiv 3 \cos 4 \theta + 5\) and deduce that $$c ^ { 6 } + s ^ { 6 } \equiv \cos ^ { 2 } 2 \theta + \frac { 1 } { 4 } \sin ^ { 2 } 2 \theta$$
   3. Prove by induction that, for all positive integers \(n\), $$c ^ { 2 n + 4 } + s ^ { 2 n + 4 } \leqslant \cos ^ { 2 } 2 \theta + \frac { 1 } { 2 ^ { n + 1 } } \sin ^ { 2 } 2 \theta$$ [You are given that the range of the function \(\mathrm { F } _ { n }\) is \(\frac { 1 } { 2 ^ { n - 1 } } \leqslant \mathrm {~F} _ { n } ( \theta ) \leqslant 1\).] {www.cie.org.uk} after the live examination series. }