Solve for X csc(x)+cot(x)=1

Problem

csc(x)+cot(x)=1

Solution

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

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

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

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

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

1+cos(x)=sin(x)

4. Square both sides of the equation to create a quadratic form.

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

1+2*cos(x)+cos2(x)=sin2(x)

5. Substitute the Pythagorean identity sin2(x)=1−cos2(x) into the equation.

1+2*cos(x)+cos2(x)=1−cos2(x)

6. Rearrange the terms into a standard quadratic equation format.

2*cos2(x)+2*cos(x)=0

7. Factor the quadratic expression.

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

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

cos(x)=0⇒x=π/2+n*π

cos(x)=−1⇒x=π+2*n*π

9. Verify the solutions in the original equation csc(x)+cot(x)=1
   If x=π+2*n*π then sin(x)=0 which makes csc(x) and cot(x) undefined.
   If x=π/2+2*n*π then csc(π/2)+cot(π/2)=1+0=1 (Valid).
   If x=(3*π)/2+2*n*π then csc((3*π)/2)+cot((3*π)/2)=−1+0=−1 (Invalid).

Final Answer

x=π/2+2*n*π

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