Tan^2x (csc^2x-1)=1

Question

Verify the identity.

Answer 1

Hi Adolfo,

Do you mean to write

 

tan²(x)(csc²(x) - 1) = 1?

 

Take the Left Hand Side LHS;

tan²(x)(csc²(x) -1) =[sin²(x)/cos²(x)][1/sin²(x) -1]

                            =[sin²(x)/cos²(x)][(1 - sin²(x))/sin²(x)]

                            =[1/cos²(x)][1-sin²(x)]    notice that the sin²(x) at the top and bottom cancel out

                            =[1/cos²(x)][cos²(x)]    because sin²(x) + cos²(x) = 1, cos²(x) = 1 - sin²(x)

                            = cos²(x)/cos²(x)

                            = 1

Left hand side = Right hand side. qed.

Answer 2

tan^2x(csc^2x-1)=1

working with the left side only...

csc^2x=cot^2+1

csc^2x-1=cot^2

substitute

tan^2x(cot2^x)

cot=1/tan

cot^2=1/tan^2

substitute

tan^2x(1/tan^2x)

tan^2x and 1/tan^2x are reciprocals of each other and their product is 1

the left side was shown to equal the right side