4. using exact values, show that ( 1+cot ^{2} \theta=csc ^{2} \theta ) for ( \theta = 45^{circ} ). 2…

4. using exact values, show that ( 1+cot ^{2} \theta=csc ^{2} \theta ) for ( \theta = 45^{circ} ). 2 marks\n5. simplify the following expression using exact values. 3 marks\n( 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\n5. simplify the following expression using exact values. 3 marks\n( cos 60^{circ}(1-sin 30^{circ})+4 \tan 45^{circ}(\tan 60^{circ}-sin 45^{circ}) )

Answer

Explanation:

Step1: Calculate left - hand side

When (\theta = 45^{\circ}), (\cot\theta=\frac{\cos\theta}{\sin\theta}), (\cot45^{\circ}=\frac{\cos45^{\circ}}{\sin45^{\circ}} = 1). Then (1+\cot^{2}45^{\circ}=1 + 1^{2}=2).

Step2: Calculate right - hand side

(\csc\theta=\frac{1}{\sin\theta}), (\csc45^{\circ}=\frac{1}{\sin45^{\circ}}=\sqrt{2}). So (\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}).