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Question: Which of the following is equivalent to \(\dfrac{{\tan n\cos ecn}}{{\sin n\sec n}}\) ? A. \(1\) ...

Which of the following is equivalent to tan⁡ncos⁡ecnsin⁡nsec⁡n\dfrac{{\tan n\cos ecn}}{{\sin n\sec n}} ?
A. 11
B. sin⁡n\sin n
C. cos⁡n\cos n
D. cot⁡n\cot n
E. cos⁡ecn\cos ecn

Explanation

Solution

To solve these types of questions you need to know the basic trigonometric ratios and their conversions from one ratio to the other. Then try and convert the given trigonometric equations into the trigonometric equations sin⁡\sin and cos⁡\cos . Then, simplify the equation to get one of the trigonometric ratios given in the options.

Formula used: The following formulae can be used to solve such types of questions:
1. sin⁡n=1cos⁡ecn\sin n = \dfrac{1}{{\cos ecn}}
2. cos⁡n=1sec⁡n\cos n = \dfrac{1}{{\sec n}}
3. tan⁡n=sin⁡ncos⁡n\tan n = \dfrac{{\sin n}}{{\cos n}}
4. cot⁡n=cos⁡nsin⁡n\cot n = \dfrac{{\cos n}}{{\sin n}}
5. sec⁡n=1cos⁡n\sec n = \dfrac{1}{{\cos n}}
6. cos⁡ecn=1sin⁡n\cos ecn = \dfrac{1}{{\sin n}}
7. tan⁡n=1cot⁡n\tan n = \dfrac{1}{{\cot n}}

Complete step-by-step solution:
Given, the equation tan⁡ncos⁡ecnsin⁡nsec⁡n\dfrac{{\tan n\cos ecn}}{{\sin n\sec n}} ,
Now, by using the below-given formulae, replace tan⁡n\tan n, cos⁡ecn\cos ecn and sec⁡n\sec n in the given equation to change the whole equation in terms of sin⁡n\sin n and cos⁡n\cos n
tan⁡n=sin⁡ncos⁡n\tan n = \dfrac{{\sin n}}{{\cos n}}
cos⁡ecn=1sin⁡n\cos ecn = \dfrac{1}{{\sin n}}
sec⁡n=1cos⁡n\sec n = \dfrac{1}{{\cos n}}
After replacing the above ratios in the given question, we get,
=sin⁡ncos⁡n×1sin⁡nsin⁡n×1cos⁡n= \dfrac{{\dfrac{{\sin n}}{{\cos n}} \times \dfrac{1}{{\sin n}}}}{{\sin n \times \dfrac{1}{{\cos n}}}}
Simplify the above equation by canceling the like terms from the denominator and the numerator, to get,
=1cos⁡nsin⁡ncos⁡n= \dfrac{{\dfrac{1}{{\cos n}}}}{{\dfrac{{\sin n}}{{\cos n}}}}
Writing the above equation more simply, we get,
=1cos⁡n÷sin⁡ncos⁡n= \dfrac{1}{{\cos n}} \div \dfrac{{\sin n}}{{\cos n}}
Simplifying the equation by multiplying 1cos⁡n\dfrac{1}{{\cos n}} by the reciprocal of sin⁡ncos⁡n\dfrac{{\sin n}}{{\cos n}} ,
=1cos⁡n×cos⁡nsin⁡n= \dfrac{1}{{\cos n}} \times \dfrac{{\cos n}}{{\sin n}}
Canceling out the like terms from the numerator and the denominator, we get,
=1sin⁡n= \dfrac{1}{{\sin n}}
Now, from the above-mentioned formula, cos⁡ecn=1sin⁡n\cos ecn = \dfrac{1}{{\sin n}} , therefore, we get the answer as,
=cos⁡ecn= \cos ecn

Therefore, after looking at all the answers given in the options, we can conclude that option (E) cos⁡ecn\cos ecn is the correct option.

Note: There are six basic trigonometric ratios which are as follows:
sin⁡e\sin e,cos⁡ine\cos ine, tan⁡gent\tan gent, cos⁡ecant\cos ecant, sec⁡ant\sec ant and cot⁡angent\cot angent. These are abbreviated as: sin⁡\sin, cos⁡\cos, tan⁡\tan, cos⁡ec\cos ec, sec⁡\sec and cot⁡\cot, respectively. These are called ratios since they can be expressed in terms of the sides of a right-angled triangle for a particular angle θ\theta.