Verify the following:

tan^2 - sin^2 = tan^2 x sin^2

Thanks.

LS = sin^2x/cos^2 - sin^2x

= (sin^2x - sin^2cos^2x)/cos^2x
= sin^2(1 - cos^2x)cos^2
= (sin^2x)(sin^2x)/cos^2x
= (tan^2x((sin^2x)
= RS

To verify the given equation:

tan^2(x) - sin^2(x) = tan^2(x) * sin^2(x)

We can start by using a trigonometric identity for the square of the tangent function:

tan^2(x) = sec^2(x) - 1

Substituting this in the equation, we get:

(sec^2(x) - 1) - sin^2(x) = (sec^2(x) - 1) * sin^2(x)

Expanding and simplifying:

sec^2(x) - 1 - sin^2(x) = sec^2(x) * sin^2(x) - sin^2(x)

Now, let's apply another trigonometric identity:

sec^2(x) = 1 + tan^2(x)

Replacing sec^2(x) in the equation:

(1 + tan^2(x)) - 1 - sin^2(x) = (1 + tan^2(x)) * sin^2(x) - sin^2(x)

Simplifying further, we get:

tan^2(x) - sin^2(x) = tan^2(x) * sin^2(x)

Hence, the equation is verified.

In this process, we used the trigonometric identities for the square of tangent and secant functions. These identities can be found in any trigonometry textbook or online resources. Remember to be careful with the signs and use the appropriate identities based on the given equations.