Simplify csc(x)^4-cot(x)^4

Problem

csc(x)−cot(x)

Solution

1. Recognize the difference of squares pattern a2−b2=(a−b)*(a+b) where a=csc(x) and b=cot(x)

csc(x)−cot(x)=(csc(x)−cot(x))*(csc(x)+cot(x))

2. Apply the Pythagorean identity csc(x)=1+cot(x) which can be rearranged to csc(x)−cot(x)=1

(csc(x)−cot(x))*(csc(x)+cot(x))=(1)*(csc(x)+cot(x))

3. Simplify the expression by removing the factor of 1.

csc(x)+cot(x)

4. Rewrite in terms of sine and cosine to see if further simplification is possible.

1/sin(x)+cos(x)/sin(x)

5. Combine the fractions over the common denominator.

(1+cos(x))/sin(x)

Final Answer

csc(x)−cot(x)=(1+cos(x))/sin(x)

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