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 5. simplify the following expression using exact values. 3 marks ( cos 60^{circ}(1-sin 30^{circ})+4 \tan 45^{circ}(\tan 60^{circ}-sin 45^{circ}) )

4. using exact values, show that ( 1+cot ^{2} \theta=csc ^{2} \theta ) for ( \theta = 45^{circ} ). 2 marks 5. simplify the following expression using exact values. 3 marks ( cos 60^{circ}(1-sin 30^{circ})+4 \tan 45^{circ}(\tan 60^{circ}-sin 45^{circ}) )

Answer

Explanation:

Step1: Calculate the left - hand side

We know that (\cot\theta=\frac{\cos\theta}{\sin\theta}), when (\theta = 45^{\circ}), (\sin45^{\circ}=\frac{\sqrt{2}}{2}), (\cos45^{\circ}=\frac{\sqrt{2}}{2}). (\cot45^{\circ}=\frac{\cos45^{\circ}}{\sin45^{\circ}} = 1) Then (1+\cot^{2}45^{\circ}=1 + 1^{2}=2)

Step2: Calculate the right - hand side

We know that (\csc\theta=\frac{1}{\sin\theta}), when (\theta = 45^{\circ}), (\sin45^{\circ}=\frac{\sqrt{2}}{2}) (\csc45^{\circ}=\frac{1}{\sin45^{\circ}}=\sqrt{2}) (\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})