Perform the indicated operation and simplify the result
Tanx(cotx-cscx)
(sinx/cosx)(cosx/sinx - 1/sinx)
= 1 - 1/cosx
= 1 - secx
To perform the given operation and simplify the result, we'll first simplify the expression within the parentheses (cotx - cscx). Then we'll substitute the simplified expression back into the original expression and simplify further.
The given expression is: tanx(cotx - cscx)
Step 1: Simplify the expression within the parentheses (cotx - cscx).
Recall the following trigonometric identities:
1. cot(x) = 1/tan(x)
2. csc(x) = 1/sin(x)
Using the above identities, we can simplify the expression as follows:
cotx - cscx = (1/tanx) - (1/sinx)
Now, we need to find a common denominator for the two fractions. The common denominator is tanx * sinx.
cotx - cscx = (sinx - tanx)/[tanx * sinx]
Step 2: Substitute the simplified expression back into the original expression and simplify further.
Substituting the simplified expression back into the original expression, we get:
tanx * [(sinx - tanx)/(tanx * sinx)]
Now, let's cancel out the common factor of tanx in the numerator and denominator:
tanx * (sinx - tanx) / sinx
This is the simplified result of the given expression:
Simplified expression = tanx * (sinx - tanx) / sinx