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If $\sin x = \frac{4}{5}$ , find \(\frac{1 + \cot^2 x}{\csc...

If $\sin x = \frac{4}{5}$ , find $\frac{1 + \cot^2 x}{\csc^2 x - 1}$ .

A.

$\frac{13}{2}$

B.

$\frac{25}{9}$

C.

$\frac{3}{13}$

D.

$\frac{4}{11}$

View answer & explanation

Correct answer is B

$\sin x = \frac{opp}{Hyp} = \frac{4}{5}$

5 $^2$ = 4 $^2$ + adj $^2$

adj $^2$ = 25 - 16 = 9

adj = $\sqrt{9}$ = 3

$\tan x = \frac{4}{3}$

$\cot x = \frac{1}{\frac{4}{3}} = \frac{3}{4}$

$\cot^2 x = (\frac{3}{4})^2 = \frac{9}{16}$

$\csc x = \frac{1}{\sin x}$

= $\frac{1}{\frac{4}{5}} = \frac{5}{4}$

$\csc^2 x = (\frac{5}{4})^2 = \frac{25}{16}$

$\therefore \frac{1 + \cot^2 x}{\csc^2 x - 1} = \frac{1 + \frac{9}{16}}{\frac{25}{16} - 1}$

= $\frac{25}{16} \div \frac{9}{16}$

= $\frac{25}{9}$

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