Question

How do you simplify $\cot x.\sec x$ ?

Answer 1

In this question, we have to simplify the given trigonometric function. When we replace the given function with some other more understandable value until it cannot be done further, the process is known as the simplification of the function. Thus by simplification of a function, we simply mean to write the function in a more understandable way. The trigonometric function given in the question is in tangent and secant form so we convert these functions into sine and cosine functions with the help of the knowledge of the trigonometric functions and then we have to simplify the given function by applying arithmetic operations like addition, subtraction, multiplication and division.

Complete step by step answer:
We know that –
$\cot x = \dfrac{1}{{\tan x}} \\\ \Rightarrow \cot x = \dfrac{1}{{\dfrac{{\sin x}}{{\cos x}}}} \\\ \Rightarrow \cot x = \dfrac{{\cos x}}{{\sin x}} \\\$
And $\sec x = \dfrac{1}{{\cos x}}$
Using these values in the given equation, we get –
$\cot x\sec x = \dfrac{{\cos x}}{{\sin x}} \times \dfrac{1}{{\cos x}} = \dfrac{1}{{\sin x}} \\\ \Rightarrow \cot x\sec x = \cos ecx \\\$
Hence, the simplified form of $\cot x\sec x$ is $\cos ecx$ .

Note: As the name suggests, trigon means triangle and ratio signifies the ratio of two sides of a right-angled triangle is known as trigonometry. Sine, cosine, tangent, secant, cosecant and cotangent functions are the functions that are contained in the trigonometric functions. Cosecant, secant and cotangent functions are the reciprocals of sine, cosine and tangent functions respectively; thus sine, cosine and tangent are the main three functions. Also tangent is equal to the ratio of the sine function and the cosine function so we can convert one trigonometric function into another easily. At last, we have obtained the function as $\cos ecx$ that is the reciprocal of the sine function, so it cannot be simplified further.