Verify the Identity

Problem

(sin(x)−cos(x))/(sec(x)−csc(x))=sin(2*x)/2

Solution

1. Rewrite the denominator using reciprocal identities sec(x)=1/cos(x) and csc(x)=1/sin(x)

(sin(x)−cos(x))/(1/cos(x)−1/sin(x))

2. Find a common denominator for the terms in the denominator of the main fraction.

(sin(x)−cos(x))/(sin(x)−cos(x))/(cos(x)*sin(x))

3. Simplify the complex fraction by multiplying the numerator by the reciprocal of the denominator.

(sin(x)−cos(x))⋅(cos(x)*sin(x))/(sin(x)−cos(x))

4. Cancel the common factor (sin(x)−cos(x)) from the numerator and denominator.

sin(x)*cos(x)

5. Apply the double angle identity sin(2*x)=2*sin(x)*cos(x) which implies sin(x)*cos(x)=sin(2*x)/2

sin(2*x)/2

Final Answer

(sin(x)−cos(x))/(sec(x)−csc(x))=sin(2*x)/2

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