Question

If $\sin \theta +{{\sin }^{2}}\theta =1$ , what is the value of ${{\cos }^{2}}\theta +{{\cos }^{4}}\theta $ ?
(a) 0
(b) $\sqrt{2}$
(c) 1
(d) 2

Answer 1

Hint: We are going to solve this by using the above relation $\sin \theta +{{\sin }^{2}}\theta =1$ and then we will use some trigonometric identities which will be given while using it and then we have to rearrange some terms to convert it in the form of $\sin \theta +{{\sin }^{2}}\theta $ .

Complete step-by-step answer:
Let’s start by solving it,
${{\cos }^{2}}\theta +{{\cos }^{4}}\theta $
${{\cos }^{2}}\theta \left( 1+{{\cos }^{2}}\theta \right)$ $$ $$
Now let’s solve the given expression to find the value of $\theta $ ,
$\sin \theta +{{\sin }^{2}}\theta =1$
$\sin \theta =1-{{\sin }^{2}}\theta $
Now $1-{{\sin }^{2}}x={{\cos }^{2}}x$
Using this we get,
$\sin \theta ={{\cos }^{2}}\theta $
Now putting the value in ${{\cos }^{2}}\theta \left( 1+{{\cos }^{2}}\theta \right)$ we get,
$\begin{align}
  & \sin \theta \left( 1+\sin \theta \right) \\
 & =\sin \theta +{{\sin }^{2}}\theta \\
\end{align}$
We have converted ${{\cos }^{2}}\theta +{{\cos }^{4}}\theta $ in the form of an expression whose value is given.
The value of this already given in the question, and hence 1 is the correct answer.
Hence, option (c) is correct.

Note: It’s always better that we check if the answer that we have got by using the above formula we is correct or not to avoid some calculation mistake and for that we need to put some values in place of $\theta $ to check whether $\sin \theta +{{\sin }^{2}}\theta ={{\cos }^{2}}\theta +{{\cos }^{4}}\theta $ is true or not. There are a bunch of trigonometric formulas that should be kept in mind while solving these questions and if we use some different set of formulas then that will be another method to solve this question, but at some point we can see that they are nearly equal.