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How do you simplify $-\left( {{\sin }^{2}}x+{{\cos }^{2}}x \right)$ ?

Last updated date: 13th Jun 2024
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Hint: To simplify the given expression $-\left( {{\sin }^{2}}x+{{\cos }^{2}}x \right)$, we are going to use the following trigonometric property which says that the addition of square of sine and square of cosine is equal to 1. The algebraic expression of this trigonometric identity is as follows: ${{\sin }^{2}}x+{{\cos }^{2}}x=1$. Using this trigonometric identity in the above expression and then simplify it further.

Complete step-by-step answer:
The trigonometric expression given in the above problem is as follows:
$-\left( {{\sin }^{2}}x+{{\cos }^{2}}x \right)$
The above expression contains sine and cosine terms and we know the trigonometric identity which has both sine and cosine terms are as follows:
${{\sin }^{2}}x+{{\cos }^{2}}x=1$
Using the above relation in $-\left( {{\sin }^{2}}x+{{\cos }^{2}}x \right)$ we get the expression written in the bracket as 1.
  & -\left( 1 \right) \\
 & =-1 \\
Hence, we have simplified the above trigonometric expression to -1.

Note: The above kind of problems which have square of cosine and sine terms is easily solved if we know the trigonometric identities involving sine and cosine just as we have shown above:
${{\sin }^{2}}x+{{\cos }^{2}}x=1$
Knowing the above trigonometric identity can directly solve the above problem.
The alternative way of solving the above expression is that instead of remembering the formula ${{\sin }^{2}}x+{{\cos }^{2}}x=1$, if you know ${{\sin }^{2}}x=1-{{\cos }^{2}}x$ then substituting ${{\sin }^{2}}x$ as $1-{{\cos }^{2}}x$ in the above expression we get,
$\Rightarrow -\left( 1-{{\cos }^{2}}x+{{\cos }^{2}}x \right)$
In the above positive and negative ${{\cos }^{2}}x$ will be cancelled out and we are left with:
  & \Rightarrow -\left( 1+0 \right) \\
 & =-1 \\
The other similar problem which can be possible is to simplify the following problem:
$-\left( {{\cos }^{2}}x-{{\sin }^{2}}x \right)$
We know the trigonometric identity involving the subtraction of square of sine from the cosine is as follows:
${{\cos }^{2}}x-{{\sin }^{2}}x=\cos 2x$
Using the above relation in $-\left( {{\cos }^{2}}x-{{\sin }^{2}}x \right)$ we get,
$-\left( \cos 2x \right)$
So, we have seen above the benefits of remembering the formulae which will save your time in the examination.