Expert: Steve Holleran - 9/16/2008

Question
Could you explain to me how to work out a question like this one...

Show that (cot^2 )/(1 + cot^2 ) = cos^2 
and hence solve (cot^2 )/(1 + cot^2 ) = 2 sin 2  for 0<<360

( = theta if it doesnt show up properly by the way)

thankyouu

Answer
Hi Tasha,

I'm going to use @ for theta, okay?

We have on the left side :   cot^2 @ / (1 + cot^2 @)

                           =cot^2 @ / csc^2 @

                           = cot^2 @ * 1/csc^2 @

                           = cos^2 @ / sin^2 @ * sin^2 @ / 1

                           = cos^2 @

Then for the equation,

               cot^2 @ / (1 + cot^2 @) = 2 sin 2@

                             cos^2 @   = 2 sin 2@

               cos^2 @ - 2 sin 2@ = 0

               cos^2 @ - 2 sin @ cos @ = 0

               cos @ (cos @ - 2 sin @) = 0

               cos @ = 0   or              cos @ - 2 sin @ = 0

                   @ = pi/2 or 3pi/2       cos @ = 2 sin @

                                           1/2 = sin @ / cos @

                                           1/2 = tan @

                                           @ = 26.56 or 333.43 deg



Hope this helps
Steve