pROVE THAT (cotx - tan x)cosx=cosec1x-2sinx
LS = (cosx/sinx- sinx/cosx)(cosx)
= (cos^2 x - sin^2 x)/(sinxcosx) (cosx)
= (cos^2 x - sin^2 x)/sinx
RS = 1/sinx - 2sinx
= (1 - 2sin^2 x)/sinx
= (sin^2 x + cos^2 x - 2sin^2 x)/sinx
= (cos^2x - sin^2 x)/sinx
= LS
To prove the equation (cot(x) - tan(x))cos(x) = csc(x) - 2sin(x), we can start with the left-hand side (LHS) and manipulate it until it matches the right-hand side (RHS) of the equation.
1. Start with the LHS: (cot(x) - tan(x))cos(x)
2. Using trigonometric identities, rewrite cot(x) and tan(x) in terms of sine (sin(x)) and cosine (cos(x)) functions:
cot(x) = cos(x) / sin(x)
tan(x) = sin(x) / cos(x)
3. Substitute these identities into the equation:
(cos(x) / sin(x) - sin(x) / cos(x))cos(x)
4. Multiply both terms by sin(x)cos(x) in order to eliminate the denominators:
cos^2(x) - sin^2(x)
5. Recall the Pythagorean identity: sin^2(x) + cos^2(x) = 1. Rearrange it to solve for cos^2(x):
cos^2(x) = 1 - sin^2(x)
6. Substitute cos^2(x) with 1 - sin^2(x) in the equation:
1 - sin^2(x) - sin^2(x)
7. Simplify by combining like terms:
1 - 2sin^2(x)
8. Recall the reciprocal trigonometric identity: csc(x) = 1 / sin(x). Substitute it into the equation:
csc(x) - 2sin(x)
9. We have now shown that the LHS is equal to the RHS, proving the equation:
(cot(x) - tan(x))cos(x) = csc(x) - 2sin(x).