Verify the Identity

Problem

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

Solution

1. Multiply the numerator and denominator of the left side by the conjugate of the denominator, which is (sin(x)+cos(x))+1

2. Expand the numerator by grouping terms as ((sin(x)+1)−cos(x))*((sin(x)+1)+cos(x)) which results in (sin(x)+1)2−cos2(x)

3. Expand the denominator as a difference of squares: (sin(x)+cos(x))2−1

4. Simplify the numerator using the identity cos2(x)=1−sin2(x)

sin2(x)+2*sin(x)+1−(1−sin2(x))

2*sin2(x)+2*sin(x)

2*sin(x)*(sin(x)+1)

5. Simplify the denominator using the identity sin2(x)+cos2(x)=1

sin2(x)+2*sin(x)*cos(x)+cos2(x)−1

1+2*sin(x)*cos(x)−1

2*sin(x)*cos(x)

6. Divide the simplified numerator by the simplified denominator and cancel the common factor 2*sin(x)

(2*sin(x)*(sin(x)+1))/(2*sin(x)*cos(x))

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

Final Answer

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

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