Solve for x csc(x)^2=6cot(x)+8

Problem

csc^2(x)=6*cot(x)+8

Solution

1. Apply the Pythagorean identity to rewrite the equation in terms of a single trigonometric function using csc^2(x)=1+cot^2(x)

1+cot^2(x)=6*cot(x)+8

2. Rearrange the equation into a standard quadratic form by subtracting 6*cot(x) and 8 from both sides.

cot^2(x)−6*cot(x)−7=0

3. Factor the quadratic expression by finding two numbers that multiply to −7 and add to −6

(cot(x)−7)*(cot(x)+1)=0

4. Set each factor to zero to find the possible values for cot(x)

cot(x)=7

cot(x)=−1

5. Solve for x by taking the arccotangent of both values, including the general solution for all real numbers where n is an integer.

x=arccot(7)+n*π

x=(3*π)/4+n*π

Final Answer

csc^2(x)=6*cot(x)+8⇒x=arccot(7)+n*π,(3*π)/4+n*π

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