Mastering Trigonometric Identities: Simplifying Expressions Using 1 + Tan^2X = Sec^2X Identity

1 + cot^2x =

csc^2x

We can start by using the identity: 1 + tan^2x = sec^2x. If we divide both sides by tan^2x, we get:

1/tan^2x + 1 = 1/cos^2x

Now, we can replace 1/tan^2x with cot^2x:

cot^2x + 1 = 1/cos^2x

Finally, we can simplify the right-hand side by using the identity: cos^2x + sin^2x = 1. If we divide both sides by sin^2x, we get:

1/sin^2x + 1/cos^2x = 1/(sin^2x cos^2x)

So, the final expression is:

cot^2x + 1 = sin^2x cos^2x

Note that we could have also solved this problem directly using the identity: cot^2x = cos^2x/sin^2x, but the method shown here is more general and can be applied to other problems that don’t have a direct identity to use.

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