How do i establish each Identity?

Question

1. (csc x)(cos x) = cot x



2. 1 + tan2(-x) = sec2x



3. cos x(tan x + cot x) = csc x



4. tan x cot x - cos2x = sin2x



5. secx -1)(sec + 1) = tan2x

Answer 1

 

Here's a couple to get you started...

1)(cscx)(cosx)=cotx

   (1/sinx)(cosx)=cotx

   (cosx)(sinx)=cotx

   (1/tanx)=cotx

       cotx=cotx

 

 

3) cosx(tanx+cotx)=cscx

    cosx([sinx/cosx]+[cosx/sinx])=cscx

           sinx +cos²x/sinx=cscx

          (sin²x/sinx)+(cos²x/sinx)=cscx

         ([sin²x+cos²x]/sinx)=cscx

                             1/sinx=cscx

                                  cscx=cscx

 

5) (secx-1)(secx+1)=tan²x

    sec²x-1=tan²x

    you can stop here because they are equal, or...

    1+tan²x=sec²x is a Pythagorean Identity

   tan²x=sec²x-1

    tan²x=tan²x

here's 4) tanxcotx-cos²x=sin²x

              tanxcotx=sin²x+cos²x

             tanx(1/tanx)=1

                       1=sin²x+cos²x is a Pythagorean Identity

 here's 2) 1+tan²(-x)=sec²x

                tan(-x)=-tan(x)

               1+(-tanx)²=sec²x

                1+tan²x=sec²x  (Pythagorean Identity)

                      sec²x=sec²x