Verify the Identity (1-sin(x))/(cos(x))=(cos(x))/(1+sin(x))

Problem

(1−sin(x))/cos(x)=cos(x)/(1+sin(x))

Solution

1. Identify the starting side of the identity. We will begin with the left-hand side (LHS).

LHS=(1−sin(x))/cos(x)

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

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

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

(1−sin2(x))/(cos(x)*(1+sin(x)))

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

cos2(x)/(cos(x)*(1+sin(x)))

5. Simplify the expression by canceling one factor of cos(x) from the numerator and denominator.

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

Final Answer

(1−sin(x))/cos(x)=cos(x)/(1+sin(x))

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