If $$\sec \theta = \frac{5}{3}$$​ and θ is an acute angle, then the value of $$\frac{(3 \tan\theta - 5 \cos \theta)}{(5 \sin \theta - 3 \sec \theta)}$$​=?

Correct option is D

Given:

​$$\sec \theta = \frac{5}{3}$$​​

Formula Used:

​$$\sec \theta = \frac{H}{B}$$​​

​$$\tan \theta = \frac{P}{B}$$​​

​$$\sin \theta = \frac{P}{H}$$​​

​$$\cos \theta = \frac{B}{H}$$​​

Solution:

​$$\sec \theta = \frac{5}{3} = \frac{H}{B}$$​​

Then H = 5,B = 3

So using Pythagoras’ Theorem

​$$H^2 = P^2 + B^2$$​​

​$$(5)^2 = (P)^2 + (3)^2$$​​

​$$P^2 = 25 - 9$$​

​$$P =\sqrt{16} =4$$​​

So:

​$$\frac{(3 \tan\theta - 5 \cos\theta)}{(5 \sin \theta - 3 \sec \theta)}$$​​

​$$=\frac{\left(3 (\frac{4}{3}) - 5 (\frac{3}{5})\right)}{\left(5 (\frac{4}{5}) - 3 (\frac{5}{3})\right)}$$​​

​$$=\frac{(4 - 3)}{(4 - 5)}$$​​

​$$= \frac{1}{(-1)}$$​​

= -1