Question

IF $\theta$ is in the first quadrant and $\cos \theta =\frac{3}{5}$ , then value of $5\tan \theta -4cosec\theta sec\theta -4\cot \theta$ is

• A. $\frac{-14}{3}$
• B. $\frac{5}{16}$
• C. $\frac{5}{-34}$
• D. $\frac{-5}{16}$

Answer 1

Answer: (A)

$\frac{-14}{3}$

Given that, in fist quadrant $\cos \theta =\frac{3}{5}$ then the value of $5\tan \theta -4cosec\theta \sec \theta -4\cot \theta$.

$\sin \theta =\sqrt{1-{\cos }^2\theta }=\sqrt{1-\frac{9}{25}}=\pm \frac{4}{5}$

In 1st quadrant values are positive.

$\sin \theta =\frac{4}{5}$

$cosec\theta =\frac{1}{\sin \theta }=\frac{5}{4}$

$\sec \theta =\frac{1}{\cos \theta }=\frac{5}{3}$

$\tan \theta =\frac{\frac{4}{5}}{\frac{3}{5}}=\frac{4}{3}$

$\cot \theta =\frac{1}{\tan \theta }=\frac{3}{4}$

Now,

$5\tan \theta -4cosec\theta \sec \theta -4\cot \theta =5\times \left(\frac{4}{3}\right)-4\times \frac{5}{4}\left(\frac{5}{3}\right)-4\left(\frac{3}{4}\right)$

$=\frac{80-100-36}{12}=\frac{-56}{12}=-\frac{14}{3}$