Question

Which of the following is equal to $\sin x\sec x$ ?
(a) $\tan x$
(b) $\cot x$
(c) $\cos x\tan x$
(d)$\cos x\cos \text{ec}x$
(e) $\cot x\cos \text{ec}x$

Answer 1

Hint : This is a very simple question from the chapter of trigonometry. As we can see, no option is in $\sin x$ or $\sec x$ . The options have one or combination of trigonometric ratios of $\tan x$ , $\cot x$ and $\cos \text{ec}x$. This means we have to use properties of trigonometric ratios and manipulate the given trigonometric expression $\sin x\sec x$ so that it matches one of the options.

Complete step-by-step answer :
To solve this question, we will define the various trigonometric ratios.
The sine or sin of an angle is defined as the ratio of the side opposite to the angle x and the hypotenuse of the right angled triangle.
Thus, $\sin x=\dfrac{opp}{hyp}$
cosine or cos of an angle is defined as the ratio of the side adjacent to the angle x and the hypotenuse of the right angled triangle.
Thus, $\cos x=\dfrac{adj}{hyp}$
tangent or tan of an angle is defined as the ratio of the side opposite to the angle x and the side adjacent to the angle x of the right angled triangle.
Thus, $\tan x=\dfrac{opp}{adj}$
$\Rightarrow \tan x=\dfrac{\sin x}{\cos x}$
cotangent or cot of an angle is defined as the ratio of the side adjacent to the angle x and the side opposite of the angle x of the right angled triangle. Thus, it is reciprocal of $\tan x$ .
Therefore, $\cot x=\dfrac{adj}{opp}$
$\Rightarrow \cot x=\dfrac{1}{\tan x}=\dfrac{\cos x}{\sin x}$
secant or sec of an angle is defined as the ratio of the hypotenuse and the side adjacent to angle x of the right angled triangle. Thus, it is reciprocal of $\cos x$ .
Therefore, $\sec x=\dfrac{hyp}{adj}$
$\Rightarrow \sec x=\dfrac{1}{\cos x}$
cosecant or cosec of an angle is defined as the ratio of the hypotenuse and the side opposite to angle x of the right angled triangle. Thus, it is reciprocal of $\sin x$ .
Therefore, $\cos \text{ec}x=\dfrac{hyp}{opp}$
$\Rightarrow \cos \text{ec}x=\dfrac{1}{\sin x}$
The expression given to us is $\sin x\sec x$ . In this expression, we will replace $\sec x$ with $\dfrac{1}{\cos x}$ .
$\Rightarrow \dfrac{\sin x}{\cos x}$
But we know that $\dfrac{\sin x}{\cos x}$ is $\tan x$ .
$\Rightarrow \dfrac{\sin x}{\cos x}=\tan x$
So, the correct answer is “Option A”.

Note : Students are advised to be well versed in the concepts of trigonometry. This is a fairly simple problem if the students know the basic properties of trigonometric ratios. The reciprocal functions must be known to solve this problem.