Question

How do you simplify \[\cot (x+y)\] to trigonometry functions of \[x\] and \[y\]?

Answer 1

Hint: We are given a trigonometric function and we have to simplify it further. We know that, \[\cot (x+y)\] function cannot be simplified further as it is. So, we will first convert it into some other trigonometric function and then simplify it further like in terms of tangent function and then use the identities of tangent function for further simplification. Therefore, we will get the simplified form of the given trigonometric function.

Complete step-by-step answer:
According to the given question, we are given a trigonometric function which we have to simplify further.
The trigonometric function we have is,
\[\cot (x+y)\]---(1)
We can see that the cotangent function cannot be further simplified in the present form. So, in order to simplify the given function, we will have to convert it into some other trigonometric form which can be easily simplified.
We know that, the inverse of cotangent function is the tangent function and the tangent function can be further simplified, so we have,
\[\cot (x+y)=\dfrac{1}{tan(x+y)}\]---(2)
And we know that tangent function can be further simplified using the identities of tangent function, so we get,
\[tan(x+y)=\dfrac{\tan x+\tan y}{1-\tan x\tan y}\]----(3)
Now, we will substitute the equation (3) in equation (2), we get,
\[\Rightarrow \cot (x+y)=\dfrac{1-\tan x\tan y}{\tan x+\tan y}\]
Therefore, we get the simplified form of the given trigonometric expression as, \[\cot (x+y)=\dfrac{1-\tan x\tan y}{\tan x+\tan y}\].

Note: The conversion of a trigonometric function to another trigonometric function should be carefully and correctly done. Also, the substitution should be done properly and appropriately. The trigonometric function to which the given function is converted should be easily simplified else it’s of no use.