Question

How do you simplify ${\sin ^2}x{\sin ^2}x$?

Answer 1

Hint: First use the rule of exponents which is given as ${a^m} \times {a^n} = {a^{m + n}}$. Since this is a trigonometric expression, we can convert it in many different forms. Use formula $\sin x = \dfrac{1}{{\csc x}}$ to get one of those forms and use ${\sin ^2}x = \dfrac{{1 - \cos \dfrac{x}{2}}}{2}$ to bring it in terms of half angles. The expression can also be converted in the form of $\cos x$ using formula ${\sin ^2}x = 1 - {\cos ^2}x$.

Complete step by step answer:
According to the question, we have been given a trigonometric expression and we are asked to simplify it.
Let the trigonometric expression be denoted by letter $y$. Then we have:
$y = {\sin ^2}x{\sin ^2}x$
First we’ll apply the rule of exponents to make it simpler. We know that if two numbers having same base are multiplied then their exponents get added as shown:
$ \Rightarrow {a^m} \times {a^n} = {a^{m + n}}$
Thus after using this rule for our trigonometric expression, we have:
$
   y = {\sin ^{2 + 2}}x \\
   \Rightarrow y = {\sin ^4}x{\text{ }}.....{\text{(1)}} \\
 $
This is the briefest simplification of the given trigonometric expression. But we can convert it in different forms.
We know the formula $\sin x = \dfrac{1}{{\csc x}}$. We can use it to convert the expression in the form of $\csc x$. So we have:
$
   y = {\sin ^4}x = {\left( {\dfrac{1}{{\csc x}}} \right)^4} \\
   \Rightarrow y = \dfrac{1}{{{{\csc }^4}x}}{\text{ }}.....{\text{(2)}} \\
 $
Another trigonometric formula we can use is:
$ \ {\sin ^2}x = \dfrac{{1 - \cos \dfrac{x}{2}}}{2}$
Using this formula, our expression will be converted into an expression of half angles. So we have:
$
   \ y = {\left( {{{\sin }^2}x} \right)^2} \\
   \Rightarrow y = {\left( {\dfrac{{1 - \cos \dfrac{x}{2}}}{2}} \right)^2} \\
   \Rightarrow y = \dfrac{1}{4}{\left( {1 - \cos \dfrac{x}{2}} \right)^2} \\
   \Rightarrow y = \dfrac{1}{4}\left( {1 + {{\cos }^2}\dfrac{x}{2} - 2\cos \dfrac{x}{2}} \right){\text{ }}.....{\text{(3)}}
 $
We can also convert the expression in the form of $\cos x$ using the formula ${\sin ^2}x = 1 - {\cos ^2}x$. Thus we have:
$
   \ y = {\left( {{{\sin }^2}x} \right)^2} \\
   \Rightarrow y = {\left( {1 - {{\cos }^2}x} \right)^2} \\
   \Rightarrow y = 1 + {\cos ^4}x - 2{\cos ^2}x{\text{ }} $

Note: The conversion of ${\sin ^4}x$ in the form of half angles in equation (3) above is widely used in integration.
$ \ {\sin ^4}x = \dfrac{1}{4}\left( {1 + {{\cos }^2}\dfrac{x}{2} - 2\cos \dfrac{x}{2}} \right)$
As the function ${\sin ^4}x$ can’t be integrated directly so we need to convert it in the form of half angles like this to do the integration.
Some other formulas used for the conversion of expressions in half angles are:
$
   \sin x = 2\sin \dfrac{x}{2}\cos \dfrac{x}{2} \\
   \Rightarrow \cos x = {\cos ^2}\dfrac{x}{2} - {\sin ^2}\dfrac{x}{2} \\
   \Rightarrow \cos x = 2{\cos ^2}\dfrac{x}{2} - 1 \\
 $