Question 438966
<pre><font face = "consolas" color = "indigo" size = 4><b>
             1             1
2csc²x = —————————  +  —————————
         1 + cos x     1 - cos x

The LCD on the right is (1 + cos x)(1 - cos x)


             1(1 - cos x)             1(1 + cos x)
2csc²x = —————————————————————— +  —————————————————————— 
         (1 + cos x)(1 - cos x)    (1 - cos x)(1 + cos x)

              1 - cos x                 1 + cos x
2csc²x = —————————————————————— +  —————————————————————— 
         (1 + cos x)(1 - cos x)    (1 - cos x)(1 + cos x)

Combine over the common denominator:

          1 - cos x  + 1 + cos x
2csc²x = ————————————————————————  
          (1 + cos x)(1 - cos x)   

Simplify the top and FOIL out the bottom

              2      
2csc²x = ———————————  
          1 - cos²x 

Use the identity {{{Sin^2theta + Cos^2theta = 1}}} written as {{{Sin^2theta = 1 - Cos^2theta}}} to 
rewrite the denominator:

            2      
2csc²x = ———————  
          sin²x


              1      
2csc²x = 2*———————  
            sin²x

Use the identity {{{Csc(theta) = 1/(Sin(theta))}}} with both sides squared
which is {{{Csc^2theta = 1/(Sin^2theta)}}} to rewrite the second
factor on the right side:

2csc²x = 2csc²x 

Edwin</pre>