Question

If tan theta equals a/b then (a sin theta -b cos theta )/(a sin theta + b cos theta) =(a2 - b2)/(a2+b2)

Answer 1

To simplify the given expression, we can use the Pythagorean identity and trigonometric identities.

Step-by-step explanation:

To solve for the given expression, we can utilize the Pythagorean identity and trigonometric identities.

From the Pythagorean identity, sin^2(theta) + cos^2(theta) = 1.

Using this identity, we can rewrite the expression as (a sin(theta) - b cos(theta))^2/(a sin(theta) + b cos(theta))^2 = (a^2 sin^2(theta) - 2ab sin(theta) cos(theta) + b^2 cos^2(theta))/(a^2 sin^2(theta) + 2ab sin(theta) cos(theta) + b^2 cos^2(theta)).

Now, we can cancel out the common terms and get (a^2 - 2ab sin(theta) cos(theta) + b^2)/(a^2 + 2ab sin(theta) cos(theta) + b^2).

Since sin(theta)/cos(theta) = tan(theta), we can rewrite the expression as (a^2 - 2ab tan(theta) + b^2)/(a^2 + 2ab tan(theta) + b^2).

By using the identity tan(a + b) = (tan(a) + tan(b))/(1 - tan(a)tan(b)), we can further simplify the expression to (a^2 - b^2)/(a^2 + b^2).