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 over the common denominator.

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

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

1+cos(x)=sin(x)

4. Square both sides to create a quadratic-style equation using the Pythagorean identity.

(1+cos(x))^2=sin^2(x)

5. Expand the left side and substitute sin^2(x)=1−cos^2(x) on the right side.

1+2*cos(x)+cos^2(x)=1−cos^2(x)

6. Rearrange the equation into standard quadratic form.

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

7. Factor the 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. 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(x)=1 and cot(x)=0 so $1 + 0 = 1(V*a*l*i*d).I*ƒ = \frac{3\pi}{2} + 2n\pi,t*h*e*ncsc(x) = -1a*n*dcot(x) = 0,s(o)1 + 0 \neq 1$ (Invalid).

Final Answer

x=π/2+2*n*π

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