SOLUTION: Hello, If {{{ cos a }}} = {{{ tan b }}} and {{{ cos b = tan a }}}, where a and b are acute angles, show that sin a = sin b = {{{ sqrt((3 - sqrt(5))/2) }}} --- I know t

Algebra ->  Trigonometry-basics -> SOLUTION: Hello, If {{{ cos a }}} = {{{ tan b }}} and {{{ cos b = tan a }}}, where a and b are acute angles, show that sin a = sin b = {{{ sqrt((3 - sqrt(5))/2) }}} --- I know t      Log On


   

Question 95547This question is from textbook
: Hello,
If +cos+a+ = +tan+b+ and +cos+b+=+tan+a+, where a and b are acute angles, show that
sin a = sin b = +sqrt%28%283+-+sqrt%285%29%29%2F2%29+
---
I know that +tan+b+=+sin+b%2Fcos+b+ and +tan+a+=+sin+a%2Fcos+a+, but I do not know where to go from here.
Thank you in advance for your help. This question is from textbook

Answer by stanbon(75887) About Me  (Show Source):
You can put this solution on YOUR website!
If +cos+a+ = +tan+b+ and +cos+b+=+tan+a+, where a and b are acute angles, show that
A)sin a = sin b
B)= +sqrt%28%283+-+sqrt%285%29%29%2F2%29+
-------------------------
Part A:
cosa = tanb
cosa = sinb/cosb
But cosb = tana
So, cosa = sinb/tana
Therefore sinb = cosa*tana
sinb = cosa
And finally, sinb = sina
==================
Part B:
So now sina = sinb
Comment:
Since a and b are acute that implies angle a = angle b
Which implies a and b are both 45 degrees
? where they got the sqrt[(3-sqrt5)/2] is beyond me?
=====================
Cheers,
Stan H.