If cos θ = 5/13, θ being an acute angle, then the value of (cos θ + 5 cot θ)/(cosec θ - cos θ)

Question

If cos θ = \(\frac{5}{13},\) θ being an acute angle, then the value of \(\frac{cos\,\theta\,+\,5\,cot\,\theta}{cosec\,\theta-cos\,\theta}\) will be

(a) \(\frac{169}{109}\)

(b) \(\frac{155}{109}\)

(c) \(\frac{385}{109}\)

(d) \(\frac{395}{109}\)

Answer 1

(c) \(\frac{385}{109}\)

Cos θ = \(\frac{5}{13}\) ⇒ sin θ = \(\sqrt{1-cos^2\,\theta}\) = \(\sqrt{1-(\frac{5}{13})^2}\)

= \(\sqrt{1-\frac{25}{169}}=\sqrt{\frac{144}{169}}=\frac{12}{13}\)

∴ cosec θ = \(\frac{1}{sin\,\theta} = \frac{12}{13}; \) cot θ = \(\frac{cos\,\theta}{sin\,\theta}=\frac{\frac{5}{13}}{\frac{12}{13}}=\frac{5}{12}\)

Now substitute the values in the exp.

\(\frac{cos\,\theta\,+\,5\,cot\,\theta}{cosec\,\theta-cos\,\theta}\)