Question

How do you simplify ${\cos ^3}x + {\sin ^2}x\cos x$ ?

Answer 1

Hint: In this question, we are given a trigonometric equation and we have been asked to simplify it. At first, you can see a ratio common in both the terms. Take that ratio common. You will get a formula in the brackets. Put that formula and you will get your required answer.

Formula used: ${\sin ^2}x + {\cos ^2}x = 1$

Complete step-by-step solution:
We are given a trigonometric equation and we have to simplify it.
$ \Rightarrow {\cos ^3}x + {\sin ^2}x\cos x$ …. (given)
Since we do not have any other lead, we will take $\cos x$ common from the two terms.
$ \Rightarrow \cos x\left( {{{\cos }^2}x + {{\sin }^2}x} \right)$
After taking $\cos x$ common, we can see the identity formed in the brackets.
Putting${\sin ^2}x + {\cos ^2}x = 1$, we get
$ \Rightarrow \cos x\left( 1 \right)$
On simply we get
$ \Rightarrow \cos x$

Hence, ${\cos ^3}x + {\sin ^2}x\cos x = \cos x$

Note: The trick used in solving such questions is that- we first look for the formulas that we can put in the equation. For example: In this question, we could have expanded ${\sin ^2}x$ or ${\cos ^3}x$ by using the formulas.
The next step involves predicting whether the formula that we are using will help us in solving the question or not. For example: In this question, if we had put the values of ${\sin ^2}x$ or ${\cos ^3}x$, we would have made the question more complicated.