Trig Proofs

1 Which of these statements is correct? Tick one box. $$\begin{aligned} & x = 2 \Rightarrow x ^ { 2 } = 4 \\ & x ^ { 2 } = 4 \Rightarrow x = 2 \\ & x ^ { 2 } = 4 \Leftrightarrow x = 2 \\ & x ^ { 2 } = 4 \Rightarrow x = - 2 \end{aligned}$$

1

1. Statement P is \(a + b = 4\).
   Statement Q is \(\quad a = 1\) and \(b = 3\).
   Which one of the following is correct? $$\mathrm { P } \Rightarrow \mathrm { Q } , \quad \mathrm { P } \Leftrightarrow \mathrm { Q } , \quad \mathrm { P } \Leftarrow \mathrm { Q }$$
2. Statement R is \(\quad x = 2\). Statement S is \(\quad x ^ { 2 } = 4\). Which one of the following is correct? $$R \Rightarrow S , \quad R \Leftrightarrow S , \quad R \Leftarrow S$$

3 In each of the following cases choose one of the statements $$P \Rightarrow Q \quad P \Leftarrow Q \quad P \Leftrightarrow Q$$ to describe the relationship between \(P\) and \(Q\).

1. \(P : y = 3 x ^ { 5 } - 4 x ^ { 2 } + 12 x\)
   \(Q : \frac { \mathrm { d } y } { \mathrm {~d} x } = 15 x ^ { 4 } - 8 x + 12\)
2. \(\quad P : x ^ { 5 } - 32 = 0\) where \(x\) is real
   \(Q : x = 2\)
3. \(\quad P : \ln y < 0\)
   \(Q : y < 1\)

11. Prove, using algebra that $$n ^ { 2 } + 1$$ is not divisible by 4 .

4 Show that the following statement is false. $$x - 5 = 0 \Leftrightarrow x ^ { 2 } = 25$$

1. A student attempts to answer the following question:

Given that \(x\) is an obtuse angle, use algebra to prove by contradiction that $$\sin x - \cos x \geqslant 1$$ The student starts the proof with: Assume that \(\sin x - \cos x < 1\) when \(x\) is an obtuse angle $$\begin{aligned} & \Rightarrow ( \sin x - \cos x ) ^ { 2 } < 1 \\ & \Rightarrow \ldots \end{aligned}$$ The start of the student's proof is reprinted below.
Complete the proof. Assume that \(\sin x - \cos x < 1\) when \(x\) is an obtuse angle $$\Rightarrow ( \sin x - \cos x ) ^ { 2 } < 1$$

1. Use proof by contradiction to prove that the curve with equation

$$y = 2 x + x ^ { 3 } + \cos x$$ has no stationary points.

7

1. Using the triangle, show that \(\sin ^ { 2 } x + \cos ^ { 2 } x = 1\).
2. Hence prove that
   \includegraphics[max width=\textwidth, alt={}]{73d1c02b-1b7b-426d-a171-c762597cfed4-2_255_501_1779_1022} \(1 + \tan ^ { 2 } x = \frac { 1 } { \cos ^ { 2 } x }\).

7

1. Prove the identity \(\frac { \sin \theta } { \sin \theta + \cos \theta } + \frac { \cos \theta } { \sin \theta - \cos \theta } \equiv \frac { \tan ^ { 2 } \theta + 1 } { \tan ^ { 2 } \theta - 1 }\).
2. Hence find the exact solutions of the equation \(\frac { \sin \theta } { \sin \theta + \cos \theta } + \frac { \cos \theta } { \sin \theta - \cos \theta } = 2\) for \(0 \leqslant \theta \leqslant \pi\).

1

1. A joke has it that army recruits used to be instructed: "If it moves, salute it. If it doesn't move, paint it." Assume that this instruction has been carried out completely in the local universe, so that everything that doesn't move has been painted.
   1. A recruit encounters something which is not painted. What should he do, and why?
   2. A recruit encounters something which is painted. Do we know what he or she should do? Justify your answer.
2. Use a truth table to prove \(( ( ( m \Rightarrow s ) \wedge ( \sim m \Rightarrow p ) ) \wedge \sim p ) \Rightarrow s\).
3. You are given the following two rules. $$\begin{aligned} & 1 \quad ( a \Rightarrow b ) \Leftrightarrow ( \sim b \Rightarrow \sim a ) \\ & 2 \quad ( x \wedge ( x \Rightarrow y ) ) \Rightarrow y \end{aligned}$$ Use Boolean algebra to prove that \(( ( ( m \Rightarrow s ) \wedge ( \sim m \Rightarrow p ) ) \wedge \sim p ) \Rightarrow s\).

12 Given that \(\arcsin x = \arccos y\), prove that \(x ^ { 2 } + y ^ { 2 } = 1\). [Hint: Let \(\arcsin x = \theta\) ] \section*{END OF QUESTION PAPER}

10. (i) Use proof by exhaustion to show that for \(n \in \mathbb { N } , n \leqslant 4\) $$( n + 1 ) ^ { 3 } > 3 ^ { n }$$ (ii) Given that \(m ^ { 3 } + 5\) is odd, use proof by contradiction to show, using algebra, that \(m\) is even.
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4. (i) Prove, by counter-example, that the statement $$\text { " } \sec ( A + B ) \equiv \sec A + \sec B , \text { for all } A \text { and } B \text { " }$$ is false.
(ii) Prove that $$\tan \theta + \cot \theta \equiv 2 \operatorname { cosec } 2 \theta , \quad \theta \neq \frac { n \pi } { 2 } , n \in \mathbb { Z }$$

1. In this question you must show detailed reasoning.
   1. Given that \(x\) and \(y\) are positive numbers such that
   $$( x - y ) ^ { 3 } > x ^ { 3 } - y ^ { 3 }$$ prove that $$y > x$$
2. Using a counter example, show that the result in part (a) is not true for all real numbers.