Simplify sec(x)+tan(x)

Problem

sec(x)+tan(x)

Solution

1. Rewrite the trigonometric functions in terms of sine and cosine using the reciprocal and quotient identities.

sec(x)=1/cos(x)

tan(x)=sin(x)/cos(x)

2. Combine the fractions since they share a common denominator.

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

3. Multiply the numerator and denominator by the conjugate of the numerator, 1−sin(x) to explore alternative forms.

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

4. Simplify the numerator using the difference of squares and the Pythagorean identity cos2(x)=1−sin2(x)

(1−sin2(x))/(cos(x)*(1−sin(x)))=cos2(x)/(cos(x)*(1−sin(x)))

5. Cancel the common factor of cos(x) from the numerator and denominator.

cos(x)/(1−sin(x))

Final Answer

sec(x)+tan(x)=(1+sin(x))/cos(x)

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