can anyone tell me if tan-1x= 1 over tan x?
No. They are different. tan^-1 (x) is a frequently used way of writing arctan x, which means the angle whose tangent is x.
tan (tan^-1 x) = x
To prove this, we can use the relationship between tangent and arctangent. Recall that the tangent function returns the ratio of the opposite side to the adjacent side of a right triangle. Conversely, the arctangent function returns the angle whose tangent is a given value.
Let's first take the expression tan(tan^-1 x). The inner function, tan^-1 x, gives us the angle whose tangent is x. Then, taking the tangent of this angle, we get a ratio.
Now, consider a right triangle where the angle A has a tangent of x.
By definition,
tan A = x.
Taking the inverse tangent (arctangent) of both sides, we get:
A = tan^-1 x.
Taking the tangent of both sides, we have:
tan A = tan(tan^-1 x).
Since the tangent function is continuous, we can simplify this to:
tan A = x.
Therefore, tan(tan^-1 x) equals x.
So, tan^-1 x is not equal to 1 over tan x. Instead, tan(tan^-1 x) is equal to x.