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Question: How do you verify the identity \[\dfrac{1}{\tan x}+\dfrac{1}{\cot x}=\tan x+\cot x\]?...

How do you verify the identity 1tan⁡x+1cot⁡x=tan⁡x+cot⁡x\dfrac{1}{\tan x}+\dfrac{1}{\cot x}=\tan x+\cot x?

Explanation

Solution

In this problem, we have to verify the given trigonometric identity. We should know some trigonometric formulas to solve these types of problems. We can substitute the formula in the left-hand side and simplify it to get the right-hand side expression and we have used several formulas in this problem to simplify and verify the given problem.

Complete step by step answer:
We know that the given trigonometric identity to be verified is,
1tan⁡x+1cot⁡x=tan⁡x+cot⁡x\dfrac{1}{\tan x}+\dfrac{1}{\cot x}=\tan x+\cot x
Now we can take the left-hand side to verify the right-hand side,
LHS = 1tan⁡x+1cot⁡x\dfrac{1}{\tan x}+\dfrac{1}{\cot x} ……. (1)
We also know that the trigonometric formula
cot⁡x=1tan⁡x,tan⁡x=1cot⁡x\cot x=\dfrac{1}{\tan x},\tan x=\dfrac{1}{\cot x} .
We can apply the above trigonometric formula in the left-hand side (1), we get
LHS = 1tan⁡x+1cot⁡x\dfrac{1}{\tan x}+\dfrac{1}{\cot x} $$$$
LHS = cot⁡x+tan⁡x\cot x+\tan x
LHS = RHS
Therefore, the given trigonometric identity 1tan⁡x+1cot⁡x=tan⁡x+cot⁡x\dfrac{1}{\tan x}+\dfrac{1}{\cot x}=\tan x+\cot x is verified.

Note:
We can also verify this problem in another method.
We know that the given trigonometric identity to be verified is,
1tan⁡x+1cot⁡x=tan⁡x+cot⁡x\dfrac{1}{\tan x}+\dfrac{1}{\cot x}=\tan x+\cot x
Now we can take the left-hand side to verify the right-hand side,
LHS = 1tan⁡x+1cot⁡x\dfrac{1}{\tan x}+\dfrac{1}{\cot x} ……. (1)
We also know that the trigonometric formula

& \tan x=\dfrac{\sin x}{\cos x}\Rightarrow \dfrac{1}{\tan x}=\dfrac{\cos x}{\sin x} \\\ & \cot x=\dfrac{\cos x}{\sin x}\Rightarrow \dfrac{1}{\cot x}=\dfrac{\sin x}{\cos x} \\\ \end{aligned}$$ We can use the above formula in (1), we get LHS = $$\dfrac{1}{\tan x}+\dfrac{1}{\cot x}$$ LHS = $$\dfrac{\cos x}{\sin x}+\dfrac{\sin x}{\cos x}$$ LHS = $$\cot x+\tan x$$ LHS = RHS. Therefore, the given trigonometric identity $$\dfrac{1}{\tan x}+\dfrac{1}{\cot x}=\tan x+\cot x$$ is verified. Students should know some trigonometric formulas, identities, properties and rules to verify or solve these types of problems. We have used several formulas in this problem to simplify and verify the given problem which should be remembered for problems to be solved.