Final answer:
The trigonometric expression cosx(cscx-secx)-cotx simplifies to -1 by using basic trigonometric identities to rewrite and then cancel terms.
Step-by-step explanation:
To simplify the given trigonometric expression cosx(cscx-secx)-cotx, we need to employ our knowledge of trigonometric identities. First, we recognize that cscx is the reciprocal of sinx, secx is the reciprocal of cosx, and cotx is the reciprocal of tanx, which is also sinx/cosx.
We can start by rewriting the expression:
- cosx(cscx - secx) - cotx
- cosx(1/sinx - 1/cosx) - (cosx/sinx)
- (cosx/sinx) - (cosx2/cosx) - (cosx/sinx)
- (cosx/sinx) - 1 - (cosx/sinx)
- The terms (cosx/sinx) cancel out, leaving us with
- -1
Hence, the simplified form of the given expression is -1.