Given: 4 sin theta = 3 cos theta

To find: The value of 5 sin theta 7 cos theta/ 7 sin theta 5 cos theta

Solution:

We are given that 4 sin theta = 3 cos theta. We can use this to find the values of sin theta and cos theta.

Dividing both sides by 4 cos theta, we get:

sin theta / cos theta = 3/4

Using the identity tan theta = sin theta / cos theta, we get:

tan theta = 3/4

Using the Pythagorean identity, we get:

sin^2 theta + cos^2 theta = 1

Substituting sin theta / cos theta = 3/4, we get:

(3/4)^2 + cos^2 theta = 1

Simplifying, we get:

cos^2 theta = 7/16

Taking the square root, we get:

cos theta = ±√(7/16)

Since sin theta / cos theta = 3/4, and cos theta can be either positive or negative, there are two possible solutions for sin theta:

sin theta = (3/4) cos theta

or

sin theta = (-3/4) cos theta

Now, we can use these values to find the value of the given expression:

5 sin theta 7 cos theta/ 7 sin theta 5 cos theta

Substituting sin theta = (3/4) cos theta, we get:

5(3/4) cos theta 7 cos theta/ 7 (3/4) cos theta 5 cos theta

Simplifying, we get:

15/4 + 7/4 / (21/4) + 5/4

= 22/4 / 26/4

= 11/13

Therefore, the value of 5 sin theta 7 cos theta/ 7 sin theta 5 cos theta is 11/13.

Answer: 11/13