prove identity cscx-(cosx)(cotx)=sinx
LS = 1/sinx - cosx(cosx/sinx)
= (1 - cos^2x)/sinx
= sin^2x/sinx
= sinx
= RS
To prove the identity csc(x) - cos(x) cot(x) = sin(x), we start with the left-hand side (LHS) and simplify it step by step until we get the right-hand side (RHS).
Step 1: Rewrite the given identity.
LHS: csc(x) - cos(x) cot(x)
RHS: sin(x)
Step 2: Rewrite csc(x) and cot(x) in terms of sin(x) and cos(x).
Using reciprocal identities:
csc(x) = 1/sin(x)
cot(x) = cos(x)/sin(x)
Now the LHS becomes:
1/sin(x) - cos(x) * (cos(x)/sin(x))
Step 3: Simplify the expression on the LHS.
To simplify, we'll find a common denominator for the two terms. The common denominator is sin(x).
LHS: (1 - cos(x) * cos(x))/sin(x)
Step 4: Simplify further.
Using the Pythagorean identity sin^2(x) + cos^2(x) = 1, we can replace sin^2(x) with (1 - cos^2(x)):
LHS: (1 - cos^2(x))/sin(x)
Using the identity sin(x) = sqrt(1 - cos^2(x)), we can rewrite the LHS as:
LHS: sin^2(x)/sin(x)
Since sin^2(x)/sin(x) is equivalent to sin(x), we have:
LHS = sin(x)
Therefore, the LHS is equal to the RHS, and the identity is proven.