Proving trigonometric identities Quiz 1

Question 1

Evaluate $${{\left[ \dfrac{1+\cos \left( \dfrac{5s}{11} \right)}{1-\cos \left( \dfrac{5s}{11} \right)} \right]}^{\frac{1}{2}}}$$

Accepted Answer

$$	ext{cosec}\dfrac{5s}{22}+\cot\dfrac{5s}{22}$$

Answer

$$	ext{cosec}\dfrac{5s}{22}+	an \dfrac{5s}{22}$$

Answer

$$\sec \dfrac{5s}{22}+	an \dfrac{5s}{22}$$

Answer

$$\sec\dfrac{5s}{22}+\cot \dfrac{5s}{22}$$

Question 2

What is $$\dfrac{{{\cos }^{2}}\left( \dfrac{5x}{2} \right)-{{\sin }^{2}}\left( \dfrac{5x}{2} \right)}{4\sin \left( \dfrac{5x}{2} \right)\cos \left( \dfrac{5x}{2} \right)}$$ equal to?

Accepted Answer

$$\dfrac{1}{2}\cot 5x$$

Answer

$$2\cot 5x$$

Answer

$$\dfrac{1}{4}\cot 5x$$

Answer

$$4\cot 5x$$

Question 3

Write the value of $$\dfrac{{{\cos }^{3}}\left( \dfrac{2}{9}z \right)-{{\sin }^{3}}\left( \dfrac{2}{9}z \right)}{\cos \left( \dfrac{2z}{9} \right)-\sin \left( \dfrac{2z}{9} \right)}$$

Accepted Answer

$$1+\sin \left( \dfrac{2z}{9} \right)\cos \left( \dfrac{2z}{9} \right)$$

Answer

$$2\cos \left( \dfrac{2z}{9} \right)\sin \left( \dfrac{2z}{9} \right)$$

Answer

$$\cos \left( \dfrac{2z}{9} \right)\sin \left( \dfrac{2z}{9} \right)$$

Answer

$$1-\sin \left( \dfrac{2z}{9} \right)\cos \left( \dfrac{2z}{9} \right)$$

Question 4

Which of the following of equivalent to $${{\left[ \dfrac{	an \left( \dfrac{2x}{5} ight)}{\sec \left( \dfrac{2x}{5} ight)}+\dfrac{\cot \left( \dfrac{2x}{5} ight)}{\operatorname{cosec}\left( \dfrac{2x}{5} ight)} ight]}^{2}}$$

Accepted Answer

$$1+\sin \left( \dfrac{4x}{5} \right)$$

Answer

$$1-\sin \left( \dfrac{4x}{5} \right)$$

Answer

$$1+\cos \left( \dfrac{4x}{5} \right)$$

Answer

$$1-\cos \left( \dfrac{4x}{5} \right)$$

Question 5

If $$y={{\left[ \dfrac{1}{{{\cos }^{2}}\left( \dfrac{6p}{7} \right)}+\dfrac{1}{{{\sin }^{2}}\left( \dfrac{6p}{7} \right)} \right]}^{2}}$$, then write the value of y.

Accepted Answer

$${{\sec }^{4}}\left( \dfrac{6p}{7} \right){\operatorname{cosec}^{4}}\left( \dfrac{6p}{7} \right)$$

Answer

$${{\sec }^{2}}\left( \dfrac{6p}{7} \right){\operatorname{cosec}^{2}}\left( \dfrac{6p}{7} \right)$$

Answer

$${{\sin }^{4}}\left( \dfrac{6p}{7} \right){{\cos }^{4}}\left( \dfrac{6p}{7} \right)$$

Answer

$${{	an }^{2}}\left( \dfrac{6p}{7} ight){{\cot }^{2}}\left( \dfrac{6p}{7} ight)$$