Demostrate the validity of tan^2theta-sin^2theta=sin^4theta sec^2theta

tan^2 Ø - sin^2 Ø = sin^4 Ø sec^2 Ø

LS = sin^2 Ø/cos^2 Ø - sin^2 Ø
= (sin^2 Ø - sin^2 Ø cos^2 Ø)/cos^2 Ø , using a common denominator.
= (sin^2 Ø - sin^2 Ø(1 - sin^2 Ø)/cos^2 Ø
= (sin^2 Ø - sin^2 Ø + sin^4 Ø)/cos^2 ‚
= sin^4 Ø sec^2 Ø
= RS

To demonstrate the validity of the equation tan^2(theta) - sin^2(theta) = sin^4(theta) sec^2(theta), we can use trigonometric identities to simplify and manipulate both sides of the equation.

First, let’s simplify the left side of the equation using the identity tan^2(theta) = sin^2(theta) / cos^2(theta):

tan^2(theta) - sin^2(theta) = sin^2(theta) / cos^2(theta) - sin^2(theta)

Next, let’s combine the fractions with a common denominator:

= (sin^2(theta) - sin^2(theta) * cos^2(theta)) / cos^2(theta)

Now, let’s simplify the numerator by factoring out sin^2(theta):

= sin^2(theta)(1 - cos^2(theta)) / cos^2(theta)

Using the Pythagorean identity sin^2(theta) + cos^2(theta) = 1, we can substitute 1 - cos^2(theta) with sin^2(theta):

= sin^4(theta) / cos^2(theta)

Using the reciprocal identity cos^2(theta) = 1 / sec^2(theta), we can rewrite the expression as:

= sin^4(theta) sec^2(theta)

Now, comparing it to the right side of the equation, we can see that tan^2(theta) - sin^2(theta) simplifies to sin^4(theta) sec^2(theta). Therefore, the equation tan^2(theta) - sin^2(theta) = sin^4(theta) sec^2(theta) is valid.

This demonstrates how we can use trigonometric identities to manipulate and simplify expressions involving trigonometric functions to prove the validity of an equation.