question_answer

If \[\cot \theta =\frac{8}{15},\] what is the value of \[\sqrt{\frac{1-\cos \theta }{1+\cos \theta }},\] where \[\theta \] is a positive acute angle?

A)  \[\frac{1}{5}\]

B)  \[\frac{2}{5}\]     

C)  \[\frac{3}{5}\]

D)  \[\frac{4}{5}\]

Correct Answer: C

Solution :

[c] \[\cot \theta =\frac{8}{15}=\frac{\text{Base}}{\text{Perpendicular}}\]

\[\therefore \] \[AC=\sqrt{{{15}^{2}}+{{8}^{2}}}\]

\[=\sqrt{225+64}=\sqrt{289}=17\]

\[\therefore \] \[\sqrt{\frac{1-\cos \theta }{1+\cos \theta }}=\sqrt{\frac{1-\frac{8}{17}}{1+\frac{8}{17}}}\]

\[=\sqrt{\frac{\frac{9}{17}}{\frac{25}{17}}}=\sqrt{\frac{9}{15}}=\frac{3}{5}\]