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Question: How do you simplify \( \sin x + \cot x\cos x \) ?...

How do you simplify sin⁡x+cot⁡xcos⁡x\sin x + \cot x\cos x ?

Explanation

Solution

Hint : In this question, we have to simplify a trigonometric function, that is, we have to convert it in a more understandable way; it is done by replacing the given values with other simpler values until it cannot be done further. In the given equation, all the functions are in sine and cosine form so cotangent has to be converted into the terms of sine and cosine function with the help of the knowledge of the trigonometric functions and then by applying the arithmetic operations like addition, subtraction, multiplication and division.

Complete step-by-step answer :
We know that –
cot⁡x=1tan⁡x ⇒cot⁡x=1sin⁡xcos⁡x ⇒cot⁡x=cos⁡xsin⁡x   \cot x = \dfrac{1}{{\tan x}} \\\ \Rightarrow \cot x = \dfrac{1}{{\dfrac{{\sin x}}{{\cos x}}}} \\\ \Rightarrow \cot x = \dfrac{{\cos x}}{{\sin x}} \;
Using this value in the given equation, we get –
sin⁡x+cos⁡xsin⁡x×cos⁡x ⇒sin⁡2x+cos⁡2xsin⁡x   \sin x + \dfrac{{\cos x}}{{\sin x}} \times \cos x \\\ \Rightarrow \dfrac{{{{\sin }^2}x + {{\cos }^2}x}}{{\sin x}} \;
We know that
sin⁡2x+cos⁡2x=1 ⇒sin⁡x+cot⁡xcos⁡x=1cos⁡x ⇒sin⁡x+cot⁡xcos⁡=sec⁡x   {\sin ^2}x + {\cos ^2}x = 1 \\\ \Rightarrow \sin x + \cot x\cos x = \dfrac{1}{{\cos x}} \\\ \Rightarrow \sin x + \cot x\cos = \sec x \;
Hence, the simplified form of cos⁡x+cot⁡xsec⁡x\cos x + \cot x\sec x is sec⁡x\sec x .
So, the correct answer is “ sec⁡x\sec x ”.

Note : Trigonometric functions include sine, cosine, tangent, secant, cosecant and cotangent functions. Trigonometric ratio signifies the ratio of two sides of a right-angled triangle. Sine, cosine and tangent are the main functions whereas cosecant, secant and cotangent functions are the reciprocals of sine, cosine and tangent functions respectively. Cotangent is the reciprocal of the tangent function and tangent is equal to the ratio of the sine function and the cosine function so we can convert cotangent function into another easily. Now, all the functions are in terms of sine and cosine functions. At last, we have obtained the function as sec⁡x\sec x that is the reciprocal of the cosine function, so it cannot be simplified further. The purpose of simplifying a function is to convert it into an easier and lesser number of terms so that they can be substituted in another bigger function.