Prove that $\sin^4(x)\cos^2(x) + \sin^2(x)\cos^4(x) - \sin^2(x)\cos^2(x)=0.$

Question

Using the following expression:

$$\sin^4(x)\cos^2(x) + \sin^2(x)\cos^4(x) - \sin^2(x)\cos^2(x).$$

The above expression is supposed to evaluate to zero, but how?

Answer 1

Hint: All three terms have a factor of $\sin^2x\cos^2x$. What happens if you take this common factor out?

Answer 2

Let $\sin^2(x) = y$.

Note that $\cos^2(x) = 1-\sin^2(x) = 1- y$.

Your expression now becomes:

$$y^2(1-y) + y(1-y)^2 - y(1-y) = y(1-y)(y + (1-y) - 1) = y(1-y)(0) = 0.$$

Answer 3

\begin{align} \sin^4 x \cos^2 x + \sin^2 x\cos^4 x - \sin^2 x \cos^2 x&=\sin^2 x \cos^2 x\ (\sin^2 x + \cos^2 x - 1)\\ &=\sin^2 x \cos^2 x\ (1-1)\\ &=\large\color{blue}0. \end{align}