How do you simplify #(2)/(1/tan x-tan x)# ?

#(2)/(1/tan x-tan x)#

1 Answer
Kevin B.
Aug 5, 2017

I'm going to guess what you mean by simplify and say that this expression simplifies down to #tan(2x)#.

Explanation:

(I am assuming that this problem is about rewriting a complicated trigonometric expression in different form; often this is done as an exercise of proving identities).

So we start with the expression
#2/(1/tan x - tan x)#

Using the identity that #1/tan x = cot x#

#2/(cot x - tan x)#

Since #cot x = cos x/sin x# and #tan x = sin x/cos x#:

#2/((cos x/sin x)-(sin x/cos x)#

We can give this a common denominator.

#2/((cos^2x-sin^2x)/(cos x sin x))#

Since dividing by a fraction is the same as multiplying by its reciprocal

#(2sinxcosx)/(cos^2x-sin^2x)#

Now if we remember the double angle formulas #sin(2x)=2sinxcosx# and #cos(2x)=cos^2x-sin^2x#

#sin(2x)/cos(2x) = tan(2x)#

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