Question

How do you solve \[\tan x + \sec x = 1\]?

Answer 1

Hint: Here we will first convert the given trigonometric ratio in terms of sine and cosine function. Then, we will solve the equation and form the condition for the equation to be true. Then the possible values of the angle will be the solution set of the given equation.

Complete step by step solution:
The given equation is \[\tan x + \sec x = 1\].
We will write the given equation in terms of sin and cos function by simply expanding the terms of the equation. Therefore, we get
\[ \Rightarrow \dfrac{{\sin x}}{{\cos x}} + \dfrac{1}{{\cos x}} = 1\]
Now we will take the common denominator and take the term in the denominator to the other side of the equation. Therefore, we get
\[ \Rightarrow \dfrac{{\sin x + 1}}{{\cos x}} = 1\]
\[ \Rightarrow \sin x + 1 = \cos x\]
Now we can write the above equation as
\[ \Rightarrow \cos x - \sin x = 1\]
 We know that the value of the sin and the cos function varies between \[ - 1\] to 1. So, for the above equation to be true the value of the cos function will be 1 and the value of the sin function will be 0. Therefore, we get
\[ \Rightarrow \cos x = 1\] and \[\sin x = 0\]
Therefore the solution will be satisfied if the angle \[x\] will be \[\left\{ {..., - 4\pi , - 2\pi ,0,2\pi ,4\pi ,...} \right\}\].

Hence the solution set will be \[\left\{ {..., - 4\pi , - 2\pi ,0,2\pi ,4\pi ,...} \right\}\].

Note:
In this question, we have used trigonometry. Trigonometry is a branch of mathematics that helps us to study the relationship between the sides and the angles of a triangle. In practical life, trigonometry is used by cartographers to make maps. It is also used by the aviation and naval industries. In fact, trigonometry is even used by Astronomers to find the distance between two stars. Hence, it has an important role to play in everyday life. The three most common trigonometric functions are the tangent function, the sine, and the cosine function. In simple terms, they are written as ‘sin’, ‘cos’, and ‘tan’. Hence, trigonometry is not just a chapter to study, in fact, it is being used in everyday life.