prove if an identity
csc (x)-cot (x)=sin (x)/1+cos(x)
you just have to use () symbols so we know what you mean.
Assuming you mean
(csc (x)-cot (x))=sin (x)/(1+cos(x))
then
cscx-cotx = 1/sinx - cosx/sinx
= (1-cosx)/sinx
now, multiply top and bottom by 1+cosx and you get
(1-cosx)(1+cosx) / (sinx(1+cosx))
= (1-cos^2x)/ (sinx(1+cosx))
and it should be easy from here on.
To prove the given identity, we'll start from the left-hand side (LHS) and manipulate it until we reach the right-hand side (RHS) of the equation. Here's the step-by-step process:
LHS: csc(x) - cot(x)
Step 1: Convert csc(x) and cot(x) to their reciprocal trigonometric functions using the following identities:
- csc(x) = 1/sin(x)
- cot(x) = cos(x)/sin(x)
LHS: 1/sin(x) - cos(x)/sin(x)
Step 2: Find the common denominator for the two fractions, which is sin(x). To do this, multiply the first fraction by sin(x)/sin(x):
LHS: (1 * sin(x))/sin(x) - cos(x)/sin(x)
LHS: sin(x)/sin(x) - cos(x)/sin(x)
Step 3: Combine the fractions by subtracting the numerators:
LHS: (sin(x) - cos(x))/sin(x)
Step 4: Simplify the expression by multiplying both numerator and denominator by the conjugate of the denominator, which is 1 + cos(x):
LHS: (sin(x) - cos(x))/sin(x) * (1 + cos(x))/(1 + cos(x))
Step 5: Apply the distributive property and multiply the numerators and denominators:
LHS: (sin(x) - cos(x))(1 + cos(x))/(sin(x)(1 + cos(x)))
Step 6: Expand the numerator using distributive property:
LHS: sin(x) + sin(x) * cos(x) - cos(x) - cos^2(x)/(sin(x)(1 + cos(x)))
Step 7: Combine like terms:
LHS: sin(x) - cos^2(x) + sin(x) * cos(x) - cos(x)/(sin(x)(1 + cos(x)))
Step 8: Rearrange the terms in the numerator:
LHS: sin(x) - cos^2(x) - cos(x) + sin(x) * cos(x)/(sin(x)(1 + cos(x)))
Step 9: Factor out common terms:
LHS: [sin(x) - cos(x)] - [cos(x) - sin(x) * cos(x)]/(sin(x)(1 + cos(x)))
Step 10: Use the identity sin^2(x) + cos^2(x) = 1 to replace sin^2(x) in the numerator:
LHS: [sin(x) - cos(x)] - [cos(x) - sin(x) * cos(x)]/(sin(x)(1 + cos(x)))
LHS: [sin(x) - cos(x)] - [cos(x) - sin^2(x)]/(sin(x)(1 + cos(x)))
LHS: [sin(x) - cos(x)] - [cos(x) - (1 - cos^2(x))]/(sin(x)(1 + cos(x)))
Step 11: Simplify the expression further:
LHS: [sin(x) - cos(x)] - [cos(x) - 1 + cos^2(x)]/(sin(x)(1 + cos(x)))
LHS: [sin(x) - cos(x)] - [cos(x) - 1 + cos^2(x)]/(sin(x)(1 + cos(x)))
LHS: [sin(x) - cos(x)] - [cos(x) - 1 + cos^2(x)]/(sin(x)(1 + cos(x)))
LHS: [sin(x) - cos(x)] - [cos(x) - 1 + cos^2(x)]/(sin(x)(1 + cos(x)))
Step 12: Simplify the expression in the square brackets:
LHS: 0 - [1 - 1 + cos^2(x)]/(sin(x)(1 + cos(x)))
LHS: -cos^2(x)/(sin(x)(1 + cos(x)))
Step 13: Simplify the numerator:
LHS: -cos^2(x)/(sin(x)(1 + cos(x))) = -(1 - sin^2(x))/(sin(x)(1 + cos(x)))
Step 14: Apply the identity 1 - sin^2(x) = cos^2(x):
LHS: -cos^2(x)/(sin(x)(1 + cos(x))) = -cos^2(x)/(sin(x)(1 + cos(x)))
RHS: sin(x)/(1 + cos(x))
As we can see, the simplified LHS is equal to the RHS, which confirms that the given identity is true.