# Prove csc^2 x + sec^2 x = csc^2 x sec^2 xRS shows the 2 sets being multiplied.

*print*Print*list*Cite

Expert Answers

hala718 | Certified Educator

csc^2 x + sec^2 x = csc^2 x * sec^2 x

We will start from the left side and prove the right side.

We know that:

csc(x) = 1/sin(x) ==> csc^2 x = 1/sin^2 x

sec(x) = 1/cos(x) ==> sec^2 x = 1/cos^2 x

We will substitute:

==> (1/sin^2 x) + (1/cos^2 x)

Now we will rewrite using the common denominator (sin^2 x*cos^2 x)

==> (cos^2 x + sin^2 x) / (sin^2 x* cos^2 x)

Now we know that sin^2 x + cos^2 x = 1

==> 1 / (sin^2 x *cos^2 x)

==> 1/sin^2 x * 1/cos^2 x

==> csc^2 x * sec^2 x ...................q.e.d