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

Correct Answer: Option B

Explanation

(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})

There is an explanation video available below.