How do I simplify cscx-cotxcosx?
usually converting to sines and cosines works, ...
cscx-cotxcosx
= 1/sinx - (cosx/sinx)(cosx)
= (1 - cos^2x)/sinx
= sin^2x/sinx
= sinx
thank you for your help
To simplify csc(x) - cot(x) * cos(x), we can use trigonometric identities.
Step 1: Recalling the definitions of trigonometric functions:
csc(x) = 1/sin(x)
cot(x) = cos(x)/sin(x)
Step 2: Substitute the values of csc(x) and cot(x) into the expression:
csc(x) - cot(x) * cos(x) = 1/sin(x) - (cos(x)/sin(x)) * cos(x)
Step 3: To add the fractions, we need a common denominator, which is sin(x):
csc(x) - cot(x) * cos(x) = 1/sin(x) - (cos^2(x))/sin(x)
Step 4: Combine the fractions by subtracting the numerators:
csc(x) - cot(x) * cos(x) = (1 - cos^2(x))/sin(x)
Step 5: Using the Pythagorean identity sin^2(x) + cos^2(x) = 1, we can rewrite the numerator:
csc(x) - cot(x) * cos(x) = sin^2(x)/sin(x)
Step 6: Simplify by canceling out sin(x) in the numerator:
csc(x) - cot(x) * cos(x) = sin(x)
Therefore, csc(x) - cot(x) * cos(x) simplifies to sin(x).
To simplify the expression csc(x) - cot(x)cos(x), we can start by rewriting the expression using trigonometric identities.
The first step is to rewrite csc(x) and cot(x) in terms of sine and cosine. Recall that csc(x) is the reciprocal of sine(x) and cot(x) is the reciprocal of tangent(x).
csc(x) = 1/sin(x)
cot(x) = 1/tan(x) = cos(x)/sin(x)
Next, substitute the rewritten values into the original expression:
csc(x) - cot(x)cos(x) = 1/sin(x) - (cos(x)/sin(x)) * cos(x)
To simplify further, we can combine the two fractions into a single fraction with a common denominator:
csc(x) - cot(x)cos(x) = (1 - cos(x) * cos(x))/sin(x)
Further simplifying the numerator, we use the trigonometric identity:
cos^2(x) + sin^2(x) = 1
Rearranging this identity, we can write:
1 - cos^2(x) = sin^2(x)
Therefore, the simplified expression becomes:
csc(x) - cot(x)cos(x) = sin^2(x)/sin(x) = sin(x)
So, the simplified expression is sin(x).
In conclusion, the simplified form of csc(x) - cot(x)cos(x) is sin(x).