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) $\cot x\cos \text{ec}x$

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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.

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.