Simplifying trigonometric expressions Quiz 1

Question 1

Simplify $$\dfrac{{{x}^{{{\sin }^{-1}}\left( \dfrac{19\ell }{11} \right)+{{\cos }^{-1}}\left( \dfrac{19\ell }{11} \right)}}}{{{x}^{{{\left[ {{\sin }^{2}}\left( \dfrac{3t}{7} \right)+{{\cos }^{2}}\left( \dfrac{3t}{7} \right) \right]}^{\frac{3}{5}}}}}}$$.

Accepted Answer

$${{x}^{\frac{\pi }{2}-1}}$$

Answer

$${{x}^{\frac{\pi }{2}}}$$

Answer

$${{x}^{\frac{\pi }{4}-1}}$$

Answer

$${{x}^{\frac{\pi }{4}}}$$

Question 2

Simplify the expression $$\dfrac{\left( \cos \dfrac{6}{7}x-\,\sin \dfrac{6}{7}x \right)\,\left( \cos \dfrac{6}{7}x+\sin \dfrac{6}{7}x \right)}{\sin \left( \dfrac{12}{7}x \right)}$$.

Accepted Answer

$$\cot \left( \dfrac{12}{7}x \right)$$

Answer

$$tan\left( \dfrac{12}{7}x \right)$$

Answer

$$\cot \left( \dfrac{6}{7}x \right)$$

Answer

$$	an \left( \dfrac{6}{7}x ight)$$

Question 3

Simplify the expression $$\dfrac{{{\sin }^{16}}\left( \dfrac{m}{4} \right)\,-\,{{\cos }^{16}}\left( \dfrac{m}{4} \right)}{{{\sin }^{8}}\left( \dfrac{m}{4} \right)\,+\,{{\cos }^{8}}\left( \dfrac{m}{4} \right)}$$.

Accepted Answer

$$-\cos \,\dfrac{m}{2}\,\,\left( {{\sin }^{4}}\dfrac{m}{4}+{{\cos }^{4}}\dfrac{m}{4} \right)$$

Answer

$$-\sin \,\dfrac{m}{2}\,\left( {{\sin }^{4}}\dfrac{m}{4}+{{\cos }^{4}}\dfrac{m}{4} \right)$$

Answer

$$\cos \dfrac{m}{2}\,\left( {{\sin }^{4}}\dfrac{m}{4}+{{\cos }^{4}}\dfrac{m}{4} \right)$$

Answer

$$\sin \dfrac{m}{2}\,\,\left( {{\sin }^{4}}\dfrac{m}{4}-{{\cos }^{4}}\dfrac{m}{4} \right)$$

Question 4

Let  $$\alpha \,=\,{{\left[ \cos \,\left( \dfrac{3g}{2} \right)+\,\sin \,\left( \dfrac{3g}{2} \right) \right]}^{2}}$$ and $$\beta \,=\,{{\left[ \cos \,\left( \dfrac{3g}{2} \right)-\,\sin \,\left( \dfrac{3g}{2} \right) \right]}^{2}}$$, then write the value  of $$\dfrac{\alpha }{\beta }$$.

Accepted Answer

$$\dfrac{1+\sin \,3g}{1-\,\sin \,3g}$$

Answer

$$\dfrac{1-\sin \,3g}{1+\,\sin \,3g}$$

Answer

$$\dfrac{1+	an \,2g}{1-\,	an \,2g}$$

Answer

$$\dfrac{1+\cot \,2g}{1-\,\cot \,2g}$$

Question 5

Which of the following is not the value of the trigonometric expressions  $${{\cos }^{2}}\left( \dfrac{23}{49}\ell  \right)-\,{{\sin }^{2}}\left( \dfrac{23}{49}\ell  \right)?$$

Accepted Answer

$$2\,\,{{\cos }^{2}}\left( \dfrac{46}{49}\ell  \right)-1$$

Answer

$$\cos \left( \dfrac{46}{49}\ell  \right)$$

Answer

$$1-2\,\,{{\sin }^{2}}\left( \dfrac{23}{49}\ell  \right)$$

Answer

$$2\,\,{{\cos }^{2}}\left( \dfrac{23}{49}\ell  \right)-1$$