Show that sec^2x-tan^2x/1+cot^2x=sinx how ?

To prove that (sec^2x - tan^2x)/(1 + cot^2x) = sinx, we will start by simplifying the left side of the equation.

Given:
(sec^2x - tan^2x)/(1 + cot^2x)

Using the trigonometric identities:
sec^2x = 1 + tan^2x
cot^2x = 1 + tan^2x

Substitute these identities into the equation:
((1 + tan^2x) - tan^2x)/(1 + (1 + tan^2x))

Simplify the numerator:
(1 + tan^2x - tan^2x)/(1 + 1 + tan^2x)
(1)/(2 + tan^2x)

Now, we will use the Pythagorean identity:
1 + tan^2x = sec^2x

Substitute this identity into the equation:
(1)/(2 + sec^2x)

Now, we will use the identity:
sec^2x = 1/cos^2x

Substitute this identity into the equation:
(1)/(2 + 1/cos^2x)
(1)/(2cos^2x + 1)

Now, we will use the identity:
1 = sin^2x + cos^2x

Substitute this identity into the equation:
(1)/(2(1 - sin^2x) + 1)
(1)/(2 - 2sin^2x + 1)
(1)/(3 - 2sin^2x)

Now, we will use the identity:
sin^2x = 1 - cos^2x

Substitute this identity into the equation:
(1)/(3 - 2(1 - cos^2x))
(1)/(3 - 2 + 2cos^2x)
(1)/(1 + 2cos^2x)

Now, we will use the identity:
cos^2x = 1 - sin^2x

Substitute this identity into the equation:
(1)/(1 + 2(1 - sin^2x))
(1)/(1 + 2 - 2sin^2x)
(1)/(3 - 2sin^2x)

Finally, we get:
(1)/(3 - 2sin^2x) = sinx

Therefore, we have proven that (sec^2x - tan^2x)/(1 + cot^2x) = sinx.