Simplify:[(cscx-cotx)(cscx+cotx)]/cscx
[(cscx-cotx)(cscx+cotx)]/cscx
expand
= (csc^2 x - cot^2 x)/csc x
= (1 + cot^2 x - cot^2 x)/csc x
= 1/csc x
= sin x
To simplify the expression [(cscx - cotx)(cscx + cotx)] / cscx, we will use the properties of trigonometric identities. Let's break it down step by step:
Step 1: Expand the numerator
The numerator is [(cscx - cotx)(cscx + cotx)]. We can expand this using the distributive property, i.e., (a - b)(a + b) = a^2 - b^2.
[(cscx - cotx)(cscx + cotx)] = (cscx)^2 - (cotx)^2
Step 2: Simplify the numerator
Using the trigonometric identities, cscx = 1/sinx and cotx = cosx/sinx, we can substitute them into the numerator.
(cscx)^2 - (cotx)^2 = (1/sinx)^2 - (cosx/sinx)^2 = (1/sinx)^2 - ((cosx)^2 / (sinx)^2)
Step 3: Simplify further
Now, we can simplify the expression further. To do that, we'll find the common denominator and combine the terms.
(1/sinx)^2 - ((cosx)^2 / (sinx)^2) = (1 - (cosx)^2) / (sinx)^2
Step 4: Reduce the fraction
We have (1 - (cosx)^2) / (sinx)^2. Notice that according to the Pythagorean identity, sin^2x + cos^2x = 1.
So, we can rewrite (1 - (cosx)^2) as sin^2x.
(sin^2x) / (sinx)^2 = sin^2x / sin^2x = 1
Step 5: Simplify the expression
Finally, we have 1 / cscx, and we know that cscx = 1/sinx.
1 / cscx = 1 / (1/sinx) = sinx
Therefore, the simplified expression is sinx.