Question

How do you simplify and write the trigonometric expression in terms of sine and cosine: for $\cos x\dfrac{\sec x}{\tan x}=f(x)$ ?

Answer 1

Hint: In this question, we have to simplify the given trigonometric function. Thus, we will use the trigonometric formula to get the required result for the solution. We will first change the secx and tanx in terms of sine and cosine functions by applying the trigonometric formula $\sec x=\dfrac{1}{\cos x}$ and $\tan x=\dfrac{\sin x}{\cos x}$ in the given trigonometric formula. After that, we will use the basic mathematical rules, to get the required solution for the problem.

Complete step by step solution:
According to the problem, we have to simplify the given trigonometric function.
Thus, we will use the trigonometric formula to get the solution.
The trigonometric function given to us is $\cos x\dfrac{\sec x}{\tan x}=f(x)$ ------- (1)
Now, we will first apply the trigonometric formula $\sec x=\dfrac{1}{\cos x}$ and $\tan x=\dfrac{\sin x}{\cos x}$ in equation (1), we get
$\Rightarrow \cos x\dfrac{\dfrac{1}{\cos x}}{\dfrac{\sin x}{\cos x}}=f(x)$
Now, we will solve the above equation furthermore by taking the reciprocal of the denominator, we get
$\Rightarrow \cos x\dfrac{1}{\cos x}.\dfrac{\cos x}{\sin x}=f(x)$
As we know, the same terms in the numerator and the denominator cancel out with a quotient 1 and remainder 0. In the above equation, we see that cosx is common in both the numerator and the denominator, thus we get
 $\Rightarrow \dfrac{\cos x}{\sin x}=f(x)$ -------- (2)
Thus, on equating equation (1) and (2), we get
$\Rightarrow \cos x\dfrac{\sec x}{\tan x}=\dfrac{\cos x}{\sin x}$ which is the required result.
Thus, for the trigonometric function $\cos x\dfrac{\sec x}{\tan x}=f(x)$ , its simplified value in terms of sine and cosine is $\dfrac{\cos x}{\sin x}$ .

Note: While solving this problem, do mention all the trigonometric formulas you are using to avoid mathematical error. Since, the answer should be in terms of sine and cosine function, thus do not end your answer by again using the trigonometric formula $\tan x=\dfrac{\sin x}{\cos x}$ , because that will leads to an inaccurate answer.