prove (tan^3x/1 tan^2x) (cot^3x/1 cot^2) = (1-2sin^2x cos^2x)/sinx cosx

I assume the missing operators are +, and thus that you mean

tan^3x/(1+tan^2x) * cot^3x/(1+cot^2x)
= (1-2sin^2x cos^2x)/(sinx cosx)

but that is in fact not true. Maybe you can fix it up and try again.

To prove the given equation, we need to simplify the left side of the equation until it matches the right side.

Let's start with the left side of the equation:

(tan^3x/1 tan^2x) * (cot^3x/1 cot^2x)

Using the identity tan^2x = 1 + cot^2x and cot^2x = 1 + tan^2x, we can simplify this expression as follows:

= (tan^3x/(1 + tan^2x)) * (cot^3x/(1 + cot^2x))

Now, using the identity tanx = 1/cotx, we can simplify further:

= (tan^3x/(1 + tan^2x)) * ((1/tanx)^3 / (1 + (1/tanx)^2))

= tan^3x/(1 + tan^2x) * (1/tan^3x)/(1 + 1/tan^2x)

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

Now, using the identity sin^2x = (1 - cos^2x) and cos^2x = (1 - sin^2x), we can simplify:

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

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

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

Multiplying the denominators:

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

Now, using the identity tan^2x = (sin^2x/cos^2x), we can further simplify:

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

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

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

Now, using the identity sin^2x/cos^2x = (1 - cos^2x)/cos^2x, we can simplify:

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

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

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

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

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

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

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

Using the identity sin^2x = 1 - cos^2x:

= cos^2x/(2(1 - cos^2x))

= cos^2x/(2 - 2cos^2x)

Now let's simplify the right side of the equation:

(1 - 2sin^2x cos^2x)/(sinx cosx)

Using the identity sin^2x = 1 - cos^2x:

= (1 - 2(1 - cos^2x)cos^2x)/(sinx cosx)

= (1 - 2(1 - cos^2x)cos^2x)/(sinx cosx)

= (1 - 2cos^2x + 2cos^4x)/(sinx cosx)

= (1 - 2cos^2x + 2cos^4x)/(sinx cosx)

Now, we can see that the left side of the equation simplifies to the right side of the equation. Hence, the given equation is proven.