Question

If \[\sin \theta + {\sin ^2}\theta = 1\], then what is the value of ${\cos ^2}\theta + {\cos ^4}\theta ?$
A. $0$
B. $\sqrt 2 $
C. $1$
D. 2

Answer 1

Hint: First of all this is a very simple and a very easy problem. In order to solve this problem we need to have some basic knowledge of trigonometry, which includes basic trigonometric identities and basic trigonometric formulas. Along with this we also need to understand and should be able to solve simple mathematical equations.
Here the trigonometric identity which is used here is as given below:
$ \Rightarrow {\sin ^2}\theta + {\cos ^2}\theta = 1$
Hence by rearranging the terms, the above expression becomes, as given below:
$ \Rightarrow {\sin ^2}\theta = 1 - {\cos ^2}\theta $

Complete step-by-step solution:
Given that \[\sin \theta + {\sin ^2}\theta = 1\], we have to find the value of ${\cos ^2}\theta + {\cos ^4}\theta $.
Consider \[\sin \theta + {\sin ^2}\theta = 1\], as given below:
$ \Rightarrow \sin \theta + {\sin ^2}\theta = 1$
$ \Rightarrow \sin \theta = 1 - {\sin ^2}\theta $
As we know that from the most important trigonometric identity that ${\sin ^2}\theta + {\cos ^2}\theta = 1$
$\therefore 1 - {\sin ^2}\theta = {\cos ^2}\theta $
Substituting the above expression in the equation $\sin \theta = 1 - {\sin ^2}\theta $, replacing $1 - {\sin ^2}\theta $with $\cos \theta $, as given below:
$ \Rightarrow \sin \theta = {\cos ^2}\theta $
Now we have found out that ${\cos ^2}\theta $ is equal to $\sin \theta $.
$\therefore {\cos ^2}\theta = \sin \theta $
Now squaring the above equation on both sides, as given below:
\[ \Rightarrow {\left( {{{\cos }^2}\theta } \right)^2} = {\left( {\sin \theta } \right)^2}\]
\[ \Rightarrow {\cos ^4}\theta = {\sin ^2}\theta \]
\[\therefore {\cos ^4}\theta = {\sin ^2}\theta \]
We obtained the expressions for both \[{\cos ^4}\theta \] and \[{\cos ^2}\theta \], which are as given below:
\[ \Rightarrow {\cos ^2}\theta = \sin \theta \] and
\[ \Rightarrow {\cos ^4}\theta = {\sin ^2}\theta \]
Substituting these above obtained expressions in the expression as given below:
\[ \Rightarrow {\cos ^2}\theta + {\cos ^4}\theta \]
\[ \Rightarrow {\cos ^2}\theta + {\cos ^4}\theta = \sin \theta + {\sin ^2}\theta \]
But already given that, \[\sin \theta + {\sin ^2}\theta = 1\]
Hence the expression will also be equal to, as given below
\[ \Rightarrow {\cos ^2}\theta + {\cos ^4}\theta = 1\]
Hence the value of ${\cos ^2}\theta + {\cos ^4}\theta = 1$

The value of ${\cos ^2}\theta + {\cos ^4}\theta $ is 1.

Note: While solving this problem we should understand that we are substituting in this equation $\sin \theta = 1 - {\sin ^2}\theta $, in place of $1 - {\sin ^2}\theta $, replacing $1 - {\sin ^2}\theta $ with $\cos \theta $. There is a chance that we might be able to confuse while substituting this. One should take care. We have to remember all the trigonometric identities such as: ${\sin ^2}\theta + {\cos ^2}\theta = 1,{\sec ^2}\theta - {\tan ^2}\theta = 1$ and $\cos e{c^2}\theta - {\cot ^2}\theta = 1$.