Simplify 1/(csc(x-1))-1/(csc(x+1))

Problem

1/csc(x−1)−1/csc(x+1)

Solution

1. Use the reciprocal identity for cosecant, which states that 1/csc(θ)=sin(θ)

1/csc(x−1)−1/csc(x+1)=sin(x−1)−sin(x+1)

2. Apply the sum-to-product identity for sine, which is sin(A)−sin(B)=2*cos((A+B)/2)*sin((A−B)/2)

A=x−1

B=x+1

3. Calculate the arguments for the cosine and sine functions in the identity.

(A+B)/2=((x−1)+(x+1))/2=(2*x)/2=x

(A−B)/2=((x−1)−(x+1))/2=(−2)/2=−1

4. Substitute the values back into the product formula.

sin(x−1)−sin(x+1)=2*cos(x)*sin(−1)

5. Apply the odd function property of sine, sin(−θ)=−sin(θ) to simplify the expression.

2*cos(x)*sin(−1)=−2*sin(1)*cos(x)

Final Answer

1/csc(x−1)−1/csc(x+1)=−2*sin(1)*cos(x)

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