SOLUTION: Prove the following is an identity. 1) 2csc^2 x = [(1/(1+cos x)] + [(1/1-cos x)] 2)cos x/(sin x*tan x+ cos x) = 1/ sec^2 x

Algebra ->  Trigonometry-basics -> SOLUTION: Prove the following is an identity. 1) 2csc^2 x = [(1/(1+cos x)] + [(1/1-cos x)] 2)cos x/(sin x*tan x+ cos x) = 1/ sec^2 x       Log On


   

Question 438966: Prove the following is an identity.
1) 2csc^2 x = [(1/(1+cos x)] + [(1/1-cos x)]
2)cos x/(sin x*tan x+ cos x) = 1/ sec^2 x

Answer by Edwin McCravy(20060) About Me  (Show Source):
You can put this solution on YOUR website!

             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%5E2theta+%2B+Cos%5E2theta+=+1 written as Sin%5E2theta+=+1+-+Cos%5E2theta to 
rewrite the denominator:

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


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

Use the identity Csc%28theta%29+=+1%2F%28Sin%28theta%29%29 with both sides squared
which is Csc%5E2theta+=+1%2F%28Sin%5E2theta%29 to rewrite the second
factor on the right side:

2csc²x = 2csc²x 

Edwin