Prove (cscx+cotx)(cscx-cotx)=1
To prove that (cscx+cotx)(cscx-cotx) equals 1, we need to simplify the expression on the left side.
Step 1: Expand the expression:
(cscx + cotx)(cscx - cotx) = cscx * cscx - cscx * cotx + cotx * cscx - cotx * cotx
Step 2: Simplify using trigonometric identities:
Recall that cscx is the reciprocal of sinx, and cotx is the reciprocal of tanx. Using these identities, we can rewrite the expression:
= (1/sinx)(1/sinx) - (1/sinx)(cosx/sinx) + (cosx/sinx)(1/sinx) - (cosx/sinx)(cosx/sinx)
= 1/(sinx * sinx) - cosx/(sinx * sinx) + cosx/(sinx * sinx) - cosx * cosx/(sinx * sinx)
Step 3: Combine like terms:
= (1 - cosx * cosx)/(sinx * sinx)
Step 4: Simplify further using the Pythagorean identity:
Recall that sinx * sinx + cosx * cosx = 1. Rearranging this equation, we have 1 - cosx * cosx = sinx * sinx.
= (sinx * sinx)/(sinx * sinx)
= 1
Therefore, we have proven that (cscx + cotx)(cscx - cotx) equals 1.