Verify the Identity cos(x)+cos(x)tan(x)^2=sec(x)

Problem

cos(x)+cos(x)*tan2(x)=sec(x)

Solution

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

cos(x)*(1+tan2(x))

2. Apply the Pythagorean identity 1+tan2(x)=sec2(x) to substitute the expression inside the parentheses.

cos(x)*sec2(x)

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

cos(x)⋅1/cos2(x)

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

1/cos(x)

5. Apply the reciprocal identity again to reach the final form.

sec(x)

Final Answer

cos(x)+cos(x)*tan2(x)=sec(x)

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