cotx-cos^(3)xcscx
what about it?
cotx - cos^3x cscx
cosx/sinx - cos^3x/sinx
cosx/sinx (1-cos^2x)
cosx sin^2x/sinx
cosx sinx
1/2 sin 2x
To simplify the expression cot(x) - cos^3(x)csc(x), we can first rewrite the trigonometric functions in terms of sine and cosine functions. Here are the steps:
Step 1: Rewrite cot(x) as cos(x)/sin(x)
cot(x) = cos(x)/sin(x)
Step 2: Rewrite csc(x) as 1/sin(x)
csc(x) = 1/sin(x)
Step 3: Substitute the rewritten expressions into the original expression:
cot(x) - cos^3(x)csc(x) = cos(x)/sin(x) - cos^3(x) * (1/sin(x))
Step 4: Combine the terms with sin(x) in the denominator:
cot(x) - cos^3(x)csc(x) = (cos(x) - cos^3(x))/sin(x)
Step 5: Factor out the common factor of cos(x):
cot(x) - cos^3(x)csc(x) = cos(x)(1 - cos^2(x))/sin(x)
Step 6: Apply the identity sin^2(x) + cos^2(x) = 1 to simplify:
cot(x) - cos^3(x)csc(x) = cos(x)(sin^2(x))/sin(x)
Step 7: Cancel out the sin(x) terms:
cot(x) - cos^3(x)csc(x) = cos(x)sin(x)
Therefore, cot(x) - cos^3(x)csc(x) simplifies to cos(x)sin(x).
To simplify the expression cot(x) - cos^3(x)csc(x), follow these steps:
Step 1: Identify trigonometric identities if possible.
- cot(x) can be rewritten as cos(x)/sin(x)
- csc(x) can be rewritten as 1/sin(x)
Step 2: Substitute the trigonometric identities into the expression.
cot(x) - cos^3(x)csc(x) = cos(x)/sin(x) - cos^3(x)(1/sin(x))
Step 3: Simplify the expression by finding a common denominator.
The common denominator is sin(x), so we need to rewrite each term with a denominator of sin(x).
cos(x)/sin(x) - cos^3(x)(1/sin(x)) = cos(x)/sin(x) - cos^3(x)/sin(x)
Step 4: Combine the terms over the common denominator.
cos(x)/sin(x) - cos^3(x)/sin(x) = (cos(x) - cos^3(x))/sin(x)
Step 5: Simplify the numerator.
The numerator can be factored as cos(x)(1 - cos^2(x)), using the identity cos^2(x) = 1 - sin^2(x).
(cos(x) - cos^3(x))/sin(x) = cos(x)(1 - cos^2(x))/sin(x)
Step 6: Apply the identity sin^2(x) + cos^2(x) = 1 to simplify the numerator further.
cos(x)(1 - cos^2(x))/sin(x) = cos(x)sin^2(x)/sin(x)
Step 7: Cancel out the common factor of sin(x) in the numerator and denominator.
cos(x)sin^2(x)/sin(x) = cos(x)sin(x)
Step 8: Simplify the expression.
The final simplified expression is cos(x)sin(x).