Question

How do you simplify \[\cos \left( -x \right)\cos x-\sin \left( -x \right)\sin x\]?

Answer 1

Hint: In this we are asked to solve the \[\cos \left( -x \right)\cos x-\sin \left( -x \right)\sin x\]expression. So for solving these type of basic trigonometric formula of \[\cos \left( -x \right)\] and \[\sin \left( -x \right)\] and we will use a trigonometric property or identity to conclude the answer and also basic arithmetic calculation to simplify further to get the final accurate and exact answer.

Complete step by step solution:
From the question, we have been given that,
\[\Rightarrow \cos \left( -x \right)\cos x-\sin \left( -x \right)\sin x\]
From the basic formula or identity of trigonometry we already know that,
\[\Rightarrow \cos \left( -\theta \right)=\cos \theta \]
And also,
\[\Rightarrow \sin \left( -\theta \right)=-\sin \theta \]
Now, we have to substitute the above mentioned property in the given question so that we can proceed further.
So, after substituting the above mentioned property in the question the expression that is in the question, we get,
\[\Rightarrow \cos \left( -x \right)\cos x-\sin \left( -x \right)\sin x\]
\[\Rightarrow \cos \left( x \right)\cos x-\left[ -\sin \left( x \right) \right]\sin x\]
Here for the above got expression we will further simplify it using arithmetic property of addition. Then, the equation will be reduced as follows.
\[\Rightarrow \cos \left( x \right)\cos x+\sin \left( x \right)\sin x\]
Here for the above got expression we will further simplify it using arithmetic property of multiplication. Then, the equation will be reduced as follows.
\[\Rightarrow {{\cos }^{2}}x+{{\sin }^{2}}x\]
Here we have to notice that the above expression after simplification is nothing but the basic identity of trigonometry. So according to the basic identity of trigonometry \[\Rightarrow {{\cos }^{2}}x+{{\sin }^{2}}x=1\] we will get the solution of the given question after using this identity as follows.
\[\Rightarrow {{\cos }^{2}}x+{{\sin }^{2}}x=1\]

Therefore, the answer for the question \[ \cos \left( -x \right)\cos x-\sin \left( -x \right)\sin x\] is 1.

Note: Students should be well aware of the basic formula of the trigonometry. Students must be very careful in doing the calculations. We must use the general property and identity of trigonometry for example in this question we used the identity \[ {{\cos }^{2}}x+{{\sin }^{2}}x=1\] so we got the solution simplified as 1. So we must be very careful in using the correct identity to the required question to get an accurate answer.