Solve for x sec(x)+tan(x)=1

Problem

sec(x)+tan(x)=1

Solution

1. Rewrite the trigonometric functions in terms of sine and cosine.

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

2. Combine the fractions on the left side since they share a common denominator.

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

3. Multiply both sides by cos(x) to eliminate the fraction, noting that cos(x)≠0

1+sin(x)=cos(x)

4. Square both sides to create a quadratic relationship, keeping in mind that this may introduce extraneous solutions.

(1+sin(x))2=cos2(x)

5. Expand the left side and use the Pythagorean identity cos2(x)=1−sin2(x) on the right side.

1+2*sin(x)+sin2(x)=1−sin2(x)

6. Rearrange the equation into a standard quadratic form by moving all terms to one side.

2*sin2(x)+2*sin(x)=0

7. Factor the quadratic expression.

2*sin(x)*(sin(x)+1)=0

8. Solve for sin(x) by setting each factor to zero.

sin(x)=0

sin(x)=−1

9. Determine the possible values for x within one period.

x=n*π

x=(3*π)/2+2*n*π

10. Verify the solutions in the original equation. If x=(3*π)/2 then cos(x)=0 which makes sec(x) and tan(x) undefined. If x=π then sec(π)+tan(π)=−1+0=−1≠1 If x=0 then sec(0)+tan(0)=1+0=1

Final Answer

sec(x)+tan(x)=1⇒x=2*n*π

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