what are the steps?

Question

what are the steps to prove that ( csc(x)+cot(x) ) ( csc(x)−cot(x) )=1

Answer 1

Hi Amanda,

 

It may look complicated but is actually quite simple if you know your trig identities

 

first you want to expand the left side so:

 

LHS (left hand side)

=(csc(x)+cot(x))(csc(x)-cot(x)

=csc²(x)+csc(x)cot(x)-csc(x)cot(x)-cot²(x)

=csc²(x)-cot²(x)

=1 (because of trig identities)

= RHS (right hand side)

 

if you don't know that csc²(x)-cot²(x)=1, then you can start with sin²(x)+cos²(x)=1, so

 

sin²(x)+cos²(x)=1 (divide both sides by sin²(x)

1+cos²(x)/sin²(x)=1/sin²(x)   (cos²(x)/sin²(x)=cot²(x) and 1/sin²(x)=csc²(x))

1+cot²(x)=csc²(x)

1=csc²(x)-cot²(x)

 

hope this helps