Disprove statement by counterexample

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

2

1. Given that \(a\) and \(b\) are real numbers, find a counterexample to disprove the statement that, if \(a > b\), then \(a ^ { 2 } > b ^ { 2 }\).
2. A student writes the statement that \(\sin x ^ { \circ } = 0.5 \Longleftrightarrow x ^ { \circ } = 30 ^ { \circ }\).
   1. Explain why this statement is incorrect.
   2. Write a corrected version of this statement.
3. Prove that the sum of four consecutive multiples of 4 is always a multiple of 8 .

1 A student states that \(1 + x ^ { 2 } < ( 1 + x ) ^ { 2 }\) for all values of \(x\).
Using a counter example, show that the student is wrong.

1. (a) Prove, by counter-example, that the statement

$$\text { "cosec } \theta - \sin \theta > 0 \text { for all values of } \theta \text { in the interval } 0 < \theta < \pi \text { " }$$ is false.
(b) Find the values of \(\theta\) in the interval \(0 < \theta < \pi\) such that $$\operatorname { cosec } \theta - \sin \theta = 2$$ giving your answers to 2 decimal places.