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How do you simplify \[\dfrac{{1 + \cos ecx}}{{\cos x + \cot x}}\] ?

Last updated date: 21st Jul 2024
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Hint: Here we have three different trigonometric functions. We will use the ratios of these functions. We will write the functions in the form of sin functions. That is cosecx and cotx has sin function in the denominator. So we will express them in sine function. And then we will solve it.

Complete step by step solution:
Given that,
\[\dfrac{{1 + \cos ecx}}{{\cos x + \cot x}}\]
Now we will write cosecx and cotx in sin function form.
\[ = \dfrac{{1 + \dfrac{1}{{\sin x}}}}{{\cos x + \dfrac{{\cos x}}{{\sin x}}}}\]
Taking the LCM in both numerator and denominator,
\[ = \dfrac{{\dfrac{{\sin x + 1}}{{\sin x}}}}{{\dfrac{{\cos x.\sin x + \cos x}}{{\sin x}}}}\]
Now cancelling the sin term,
\[ = \dfrac{{\sin x + 1}}{{\cos x.\sin x + \cos x}}\]
Taking cosx common from the denominator,
\[ = \dfrac{{\sin x + 1}}{{\cos x\left( {\sin x + 1} \right)}}\]
Cancelling the common term,
\[ = \dfrac{1}{{\cos x}}\]
We know that reciprocal of cosx is secx,
\[ = \sec x\]
Thus the answer is \[\dfrac{{1 + \cos ecx}}{{\cos x + \cot x}} = \sec x\]
So, the correct answer is “\[ \sec x\] ”.

Note: Note that, in these types of problems we use the trigonometric functions and their identities as per the need of the problem. Always try to write the equations in such a way that they can be simplified in an easy way. Like in the problem above we have taken help of sin function.