Solve for x cos(x)+sin(x)tan(x)=sec(x)

Problem

cos(x)+sin(x)*tan(x)=sec(x)

Solution

1. Rewrite the tangent function in terms of sine and cosine using the identity tan(x)=sin(x)/cos(x)

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

2. Multiply the terms in the second part of the expression.

cos(x)+sin2(x)/cos(x)=sec(x)

3. Find a common denominator for the terms on the left side by multiplying cos(x) by cos(x)/cos(x)

cos2(x)/cos(x)+sin2(x)/cos(x)=sec(x)

4. Combine the fractions over the common denominator.

(cos2(x)+sin2(x))/cos(x)=sec(x)

5. Apply the Pythagorean identity cos2(x)+sin2(x)=1 to simplify the numerator.

1/cos(x)=sec(x)

6. Apply the reciprocal identity sec(x)=1/cos(x) to the left side.

sec(x)=sec(x)

7. Identify the solution set. Since the equation simplifies to an identity, it is true for all values of x where the functions are defined. The functions tan(x) and sec(x) are undefined when cos(x)=0

cos(x)≠0

x≠π/2+n*π

Final Answer

cos(x)+sin(x)*tan(x)=sec(x)⇒x∈ℝ,x≠π/2+n*π,n∈ℤ

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