Proving identities: tell wether or not f(x)=sin x is an identity (Does it simplify to Sin x) ? Please help

(sin^3x)(1+cot^2x)

To determine whether or not the expression (sin^3x)(1+cot^2x) simplifies to sin x, we'll need to simplify the expression and check if it matches sin x.

Let's start by expanding the expression:
(sin^3x)(1+cot^2x) = sin^3x + sin^3x*cot^2x

Next, we can use the trigonometric identity for cotangent, which states that cot^2x = 1/tan^2x. Substituting this in the expression, we get:
sin^3x + sin^3x*(1/tan^2x)

To simplify further, we can use another trigonometric identity, which states that sin^2x = 1 - cos^2x. By rearranging, we have cos^2x = 1 - sin^2x. Substituting this identity, the expression becomes:
sin^3x + sin^3x*(1/(1 - sin^2x))

Next, we can simplify the expression by finding a common denominator. The denominator on the right side can be multiplied by (1 + sin^2x)/(1 + sin^2x), which gives us:
sin^3x + sin^3x*(1 + sin^2x)/(1 - sin^2x)

Now, we can simplify further by combining the terms under a common denominator:
(sin^3x*(1 - sin^2x) + sin^3x*(1 + sin^2x))/(1 - sin^2x)

Expanding the numerator, we have:
(sin^3x - sin^5x + sin^3x + sin^5x)/(1 - sin^2x)

By canceling out the like terms, the numerator simplifies to:
2sin^3x/(1 - sin^2x)

Now, we can use the trigonometric identity sin^2x = 1 - cos^2x, which gives us:
2sin^3x/(1 - (1 - cos^2x))

Simplifying further, we have:
2sin^3x/(1 - 1 + cos^2x)

The terms (1 - 1) in the denominator cancel out, leaving us with:
2sin^3x/cos^2x

Finally, we can use the identity sin^2x/cos^2x = tan^2x, to rewrite the expression as:
2sin^3x * tan^2x

After simplifying, we can see that the original expression (sin^3x)(1+cot^2x) does not simplify to sin x. Instead, it simplifies to 2sin^3x * tan^2x.