Answer: Starting with the expression:
tan(x)(cot(x) - cos(x))
Recall that cot(x) is the reciprocal of tan(x), so we can substitute cot(x) = 1/tan(x):
tan(x)(1/tan(x) - cos(x))
Simplifying:
1 - cos(x)tan(x)
Next, we can use the identity cos(x) = 1/sec(x) to eliminate the cotangent term:
1 - (1/sec(x))tan(x)
Now, we can use the identity tan(x) = sin(x)/cos(x) and sec(x) = 1/cos(x) to express the expression in terms of sine and cosine:
1 - (1/cos(x))(sin(x)/cos(x))
Simplifying:
cos(x)/cos(x) - sin(x)/cos(x)
= (cos(x) - sin(x))/cos(x)
Therefore, the simplified expression in terms of sine and cosine is:
(cos(x) - sin(x))/cos(x)
Explanation: