Verify the Identity sec(x)-sec(x)sin(x)^2=cos(x)

Problem

sec(x)−sec(x)*sin^2(x)=cos(x)

Solution

1. Factor out the common term sec(x) from the left side of the equation.

sec(x)*(1−sin^2(x))

2. Apply the Pythagorean identity sin^2(x)+cos^2(x)=1 which implies 1−sin^2(x)=cos^2(x)

sec(x)*cos^2(x)

3. Substitute the reciprocal identity sec(x)=1/cos(x) into the expression.

1/cos(x)⋅cos^2(x)

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

cos(x)

Final Answer

sec(x)−sec(x)*sin^2(x)=cos(x)

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