(sec^2x-1)(csc^2x-1)=1 prove the following identity
ive been stuck on this for hours please help
sec^2(x)-1 = tan^2(x)
csc^2(x)-1 = cot^2(x)
since cot = 1/tan, it follows immediately
A math tutor told me to FOIL it. This is just a hint, or starting point because I don't know for sure.
To prove the given identity, we need to manipulate one side of the equation to arrive at the other side. Let's start by simplifying the left-hand side (LHS) of the equation:
LHS = (sec^2x - 1)(csc^2x - 1)
Recall the trigonometric identities:
sec^2x = 1 + tan^2x
csc^2x = 1 + cot^2x
Using these identities, we can rewrite the LHS as:
LHS = (1 + tan^2x - 1)(1 + cot^2x - 1)
Simplifying further:
LHS = tan^2x * cot^2x
Now, we can rewrite the right-hand side (RHS) as follows:
RHS = 1
Since 1 is the same as tan^2x * cot^2x, we have:
LHS = RHS
Thus, the identity is proved:
(tan^2x * cot^2x) = 1
Please note that when solving trigonometric identities, it is crucial to apply known trigonometric identities and manipulate the expressions until the two sides of the equation are equal.