Solve for x cos(x)+sin(x)tan(x)=2

Problem

cos(x)+sin(x)*tan(x)=2

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)=2

2. Multiply the terms to simplify the expression.

cos(x)+sin2(x)/cos(x)=2

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)=2

4. Combine the fractions over the common denominator.

(cos2(x)+sin2(x))/cos(x)=2

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

1/cos(x)=2

6. Solve for cosine by taking the reciprocal of both sides.

cos(x)=1/2

7. Determine the general solution for x based on the unit circle values where the cosine is 1/2

x=±π/3+2*n*π

Final Answer

cos(x)+sin(x)*tan(x)=2⇒x=±π/3+2*n*π

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