4. using exact values, show that ( 1+cot ^{2} \theta=csc ^{2} \theta ) for ( \theta=45^{circ} ). 2 marks
4. using exact values, show that ( 1+cot ^{2} \theta=csc ^{2} \theta ) for ( \theta=45^{circ} ). 2 marks
Answer
Explanation:
Step1: Calculate the left - hand side
We know that (\cot\theta=\frac{\cos\theta}{\sin\theta}). For (\theta = 45^{\circ}), (\sin45^{\circ}=\frac{\sqrt{2}}{2}) and (\cos45^{\circ}=\frac{\sqrt{2}}{2}). Then (\cot45^{\circ}=\frac{\cos45^{\circ}}{\sin45^{\circ}} = 1). So (1+\cot^{2}45^{\circ}=1 + 1^{2}=2).
Step2: Calculate the right - hand side
We know that (\csc\theta=\frac{1}{\sin\theta}). For (\theta = 45^{\circ}), (\sin45^{\circ}=\frac{\sqrt{2}}{2}), then (\csc45^{\circ}=\frac{1}{\sin45^{\circ}}=\sqrt{2}). And (\csc^{2}45^{\circ}=(\sqrt{2})^{2}=2).
Answer:
Since (1+\cot^{2}45^{\circ}=2) and (\csc^{2}45^{\circ}=2), we have shown that (1 + \cot^{2}\theta=\csc^{2}\theta) for (\theta = 45^{\circ}).