Question

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

Answer 1

We will first start with the equation given as $\sin \theta + {\sin ^2}\theta = 1$. We will use trigonometric identities such as ${\sin ^2}\theta + {\cos ^2}\theta = 1$ and then we will come to a conclusion by proving the required equation simultaneously.

Complete step-by-step answer:
Let us consider the equation $\sin \theta + {\sin ^2}\theta = 1$.
We have a trigonometric identity ${\sin ^2}\theta + {\cos ^2}\theta = 1$. We can re – write this equation as
$\Rightarrow {\sin ^2}\theta = 1 - {\cos ^2}\theta$
Therefore, substituting this value of ${\sin ^2}\theta$in the equation $\sin \theta + {\sin ^2}\theta = 1$, we get
$\Rightarrow \sin \theta + 1 - {\cos ^2}\theta = 1 \\\ \Rightarrow \sin \theta - {\cos ^2}\theta = 0 \\\ \Rightarrow \sin \theta = {\cos ^2}\theta \\\$
Now, on squaring both sides of this equation, we get
$\Rightarrow {\sin ^2}\theta = {\cos ^4}\theta$
Again, using the trigonometric identity ${\sin ^2}\theta + {\cos ^2}\theta = 1$ $\Rightarrow {\sin ^2}\theta = 1 - {\cos ^2}\theta$in the above equation, we get
$\Rightarrow 1 - {\cos ^2}\theta = {\cos ^4}\theta$
Rearranging the terms of the equation, we get
$\Rightarrow {\cos ^2}\theta + {\cos ^4}\theta = 1$
Hence, we have proved the required equation ${\cos ^2}\theta + {\cos ^4}\theta = 1$by using the given equation $\sin \theta + {\sin ^2}\theta = 1$.

Note: In such problems, you may get confused about using the trigonometric identities or trigonometric relations. You can also solve this equation by considering only the left hand side of the equation ${\cos ^2}\theta + {\cos ^4}\theta = 1$ and then taking the term ${\cos ^2}\theta$ as common and then using the identity ${\sin ^2}\theta + {\cos ^2}\theta = 1$and the given equation to prove it equal to 1. This way is a bit more complex compared to the one we have solved above.