Verify the Identity 1/(1-sin(x))+1/(1+sin(x))=2sec(x)^2

Problem

1/(1−sin(x))+1/(1+sin(x))=2*sec2(x)

Solution

1. Find a common denominator for the fractions on the left side of the equation by multiplying the denominators (1−sin(x)) and (1+sin(x))

(1*(1+sin(x))+1*(1−sin(x)))/((1−sin(x))*(1+sin(x)))

2. Simplify the numerator by combining like terms, noting that sin(x) and −sin(x) cancel out.

2/((1−sin(x))*(1+sin(x)))

3. Expand the denominator using the difference of squares formula (a−b)*(a+b)=a2−b2

2/(1−sin2(x))

4. Apply the Pythagorean identity cos2(x)+sin2(x)=1 which implies 1−sin2(x)=cos2(x)

2/cos2(x)

5. Use the reciprocal identity sec(x)=1/cos(x) to rewrite the expression in terms of secant.

2*sec2(x)

Final Answer

1/(1−sin(x))+1/(1+sin(x))=2*sec2(x)

Want more problems? Check here!