(1/tan x) + (tan x)= (1/ sin x cos x)

Question

I am to prove this identity to be correct. And I don't know what side to begin with...since I started with the left and made tan into sin/cos.

 

Answer 1

Remember: tan(x) = sin(x) / cos(x)  and  1/tan(x) = cos(x) / sin(x)  and  sin^2(x) + cos^2(x)= 1

on the left side:

(cos(x) / sin(x)) + sin(x) / cos(x)

= [sin^2(x) + cos^2(x)] / sin(x)cos(x) = right side = 1/sin(x)cos(x)

Answer 2

Hi Karia,

It is always a good idea to start with the side that looks more complex.

1/tan x  + tan x = 1/(sin x / cos x) + tan x

                        = cos x / sin x   +  sin x / cos x  We need to get a common denominator. Multiply the first fraction by cos x / cos x and the second fraction by sin x / sin x to get

                          cos² x / (sin x  cos x) + sin² x / (sin x  cos x) Now we can add the fractions together and get

                          cos² x  + sin² x / sin x  cos x = 1/ sinx cos x

Answer 3

I would work out the left side, since that side is the less simplified part.

 

You know that            tanx = sinx / cosx

 

 

[1 / (sinx / cosx)] + (sinx / cosx) =

 

(cosx / sinx) + (sinx / cosx)

 

 

Now we look for the common denominator.  The common denominator is sinx cosx.

 

 

(cos² x + sin² x ) / (sinx cosx)

 

 

The numerator part is Pythagorean theorem identity that is equal to 1.

 

1 / (sinx cosx)