Proving The Trigonometric Identity: 1 + Tan^2X = Sec^2X Using Definitive Functions

1 + cot^2x =

csc^2x

One possible way to solve this equation is by using trigonometric identities. In particular, we can use the Pythagorean identity for tangent and cotangent:

1 + tan^2x = sec^2x
1 + cot^2x = csc^2x

Starting with the left-hand side of the equation, we can substitute cotangent for tangent:

1 + cot^2x = 1 + (1/tan^2x)
= (tan^2x + 1)/tan^2x
= (1 + tan^2x)/tan^2x

Then, we can use the Pythagorean identity for tangent:

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

Finally, we can simplify by using the reciprocal identity for tangent:

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

Therefore, we have shown that 1 + cot^2x = csc^2x, which is the desired result.

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