Question 800722
<pre>
Prove that tan(10°)cos(20°)tan(40°)tan(50°)csc(70°)sec(80°)sin(80°) = 1

Change all trig ratios other than the sines and cosines
to sines and cosines, and put 1 denominators under the 
sines and cosines, using these quotient and reciprocal 
identities: 

{{{tan(theta))}}}{{{""=""}}}{{{sin(theta)/cos(theta)}}}, {{{sec(theta)}}}{{{""=""}}}{{{1/cos(theta)}}}, {{{csc(theta)}}}{{{""=""}}}{{{1/sin(theta)}}}

{{{sin("10°")/cos("10°")}}}{{{""*""}}}{{{cos("20°")/1}}}{{{""*""}}}{{{sin("40°")/cos("40°")}}}{{{""*""}}}{{{sin("50°")/cos("50°")}}}{{{""*""}}}{{{1/sin("70°")}}}{{{""*""}}}{{{1/cos("80°")}}}{{{""*""}}}{{{sin("80°")/1}}}{{{""*""}}}

Next change all the cosines to sines using the cofuction identity:

{{{cos(theta)}}}{{{""=""}}}{{{sin("90°"-theta)}}}

{{{cos("10°")}}}{{{""=""}}}{{{sin("90°"-"10°")}}}{{{""=""}}}{{{sin("80°")}}}
{{{cos("20°")}}}{{{""=""}}}{{{sin("90°"-"20°")}}}{{{""=""}}}{{{sin("70°")}}}
{{{cos("40°")}}}{{{""=""}}}{{{sin("90°"-"40°")}}}{{{""=""}}}{{{sin("50°")}}}
{{{cos("50°")}}}{{{""=""}}}{{{sin("90°"-"50°")}}}{{{""=""}}}{{{sin("40°")}}}
{{{cos("80°")}}}{{{""=""}}}{{{sin("90°"-"80°")}}}{{{""=""}}}{{{sin("10°")}}}

Substitute those for all the cosines:

{{{sin("10°")/sin("80°")}}}{{{""*""}}}{{{sin("70°")/1}}}{{{""*""}}}{{{sin("40°")/sin("50°")}}}{{{""*""}}}{{{sin("50°")/sin("40°")}}}{{{""*""}}}{{{1/sin("70°")}}}{{{""*""}}}{{{1/sin("10°")}}}{{{""*""}}}{{{sin("80°")/1}}}{{{""*""}}}

That all cancels and gives 1

Edwin</pre>