Question

How do you simplify the expression $\cos x\left( {\sec x - \cos x} \right)$?

Answer 1

Hint: In this question, we have to simplify the trigonometric expression to its reduced form. For this, we have to know the trigonometric formulas to simplify these types of questions. First, apply the trigonometric reciprocal ratio of$\sec x$. Then find the least common factor of the denominator. Then apply the basic Pythagorean trigonometric formula ${\sin ^2}x + {\cos ^2}x = 1$.

Complete step by step solution:
In this question, we want to simplify the trigonometric expression,
$ \Rightarrow \cos x\left( {\sec x - \cos x} \right)$...(1)
Now, we know that the trigonometric reciprocal ratio of $\sec x$ into cosine.
That is $\sec x = \dfrac{1}{{\cos x}}$.
 Substitute the value of $\sec x$ in the equation (1).
Therefore,
$ \Rightarrow \cos x\left( {\dfrac{1}{{\cos x}} - \cos x} \right)$
Now, take the least common multiple (LCM) of the bracket.
$ \Rightarrow \cos x\left( {\dfrac{{1 - \left( {\cos x \times \cos x} \right)}}{{\cos x}}} \right)$
That is equal to,
 $ \Rightarrow \cos x\left( {\dfrac{{1 - \left( {{{\cos }^2}x} \right)}}{{\cos x}}} \right)$
Here, we can cancel $\cos x$ from the numerator and the denominator as both are the same terms.
$ \Rightarrow 1 - {\cos ^2}x$
Now, we already know the basic Pythagorean trigonometric formula ${\sin ^2}x + {\cos ^2}x = 1$ . So, we can rewrite this identity as $1 - {\cos ^2}x = {\sin ^2}x$.
Substitute this identity in the above expression.
$ \Rightarrow {\sin ^2}x$

Hence, the solution of the above equation is ${\sin ^2}x$.

Note:
We should know the basic trigonometric rules and formulas to solve these types of questions.
The reciprocal ratios of the trigonometric ratios are:
$\cos ecx = \dfrac{1}{{\sin x}}$
$\sec x = \dfrac{1}{{\cos x}}$
$\cot x = \dfrac{1}{{\tan x}}$
It can also be written as,
$\cot x = \dfrac{{\cos x}}{{\sin x}}$
The Pythagorean trigonometric identity is also known as Pythagorean identity. It is an identity expressing the Pythagorean theorem in terms of trigonometric functions along with the sine and cosine functions.
The identity is based on the sine and the cosine functions are as below.
${\sin ^2}x + {\cos ^2}x = 1$
There are other two identities which are called Pythagorean trigonometric identity :
The trigonometric identity involving the tangent and the secant functions follows from the Pythagorean theorem.
${\tan ^2}x = 1 + {\sec ^2}x$
The trigonometric identity involving the cotangent and the cosecant functions follows from the Pythagorean theorem.
${\cot ^2}x = 1 + \cos e{c^2}x$