Question

If $\cos x + {\cos ^2}x = 1$ then prove that ${\sin ^2}x + {\sin ^4}x = 1$.

Answer 1

In this geometrical trigonometry problem we will just use one identity that is ${\sin ^2}x + {\cos ^2}x = 1$. We will use this quantity to prove the statement.

Complete step-by-step answer:
Given that,
$\cos x + {\cos ^2}x = 1$
Rearranging the terms,
$\Rightarrow \cos x = 1 - {\cos ^2}x$
$\Rightarrow \cos x = {\sin ^2}x....... \to ({\sin ^2}x + {\cos ^2}x = 1)$
Then squaring both sides,
$\Rightarrow {\cos ^2}x = {\sin ^4}x........ \to equation1$
Also a simple logic is ,
${\cos ^2}x - {\cos ^2}x = 0$
Now using equation1 we will replace first term and we will keep second term as it is
${\sin ^4}x - {\cos ^2}x = 0$
Adding 1 on both sides,
${\sin ^4}x - {\cos ^2}x + 1 = 0 + 1$
${\sin ^4}x + 1 - {\cos ^2}x = 1$
${\sin ^4}x + {\sin ^2}x = 1....... \to ({\sin ^2}x + {\cos ^2}x = 1)$
Hence proved the statement.

Note: Here the point that we should understand is a trigonometric problem need not to involve too many identities it can be solved with a single quantity. Also note that we have replaced the second ${\cos ^2}x$ term because the sin term is positive in proving that part. Always start to prove that with given data.