Question

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

Answer 1

Hint: 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.