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

Question

what are the steps to verify this trig identity

Answer 1

Hi Abby,

 

LHS= tan²x(cosec² x-1)

      = sin²x /cos²x( 1/sin²x -1)..... using tan² x=sin²x/cos²x & cosec²x=1/sin² x     

       =sin²x/cos²x( 1-sin²x / sin² x)

       =1/cos²x (1-sin²x)

 

        =1/cos²x( cos²x)........   using identity, sin²x+ cos²x=1              

        = 1

        = RHS

It is proved.

 

Meena from Strongsville,OH

Answer 2

tan²x(csc²x-1)=1

 

One trig identity we are given is (1+cot²x)=csc²x.

We can rewrite this by subtracting 1 from both sides.  This gives us cot²x=csc²x-1

Notice from your equation, csc²x-1 is a part of it.

We can substitute cot²x in for csc²x-1.

 

tan²x(cot²x)=1

 

(sin²x/cos²x)×(cos²x/sin²x)=1

 

We can simplify by dividing.

sin²x/sin²x=1

cos²x/cos²x=1

 

(1/1)×(1/1)=1