Question

Question Solve the equation on the interval 0 ≤ θ < 2π.
cot θ = csc θ -1

Answer 1

\[\cot{x} = \frac{\cos{x}}{\sin{x}};\; \csc{x} = \frac{1}{\sin{x}}\]
\[\frac{\cos{x}}{\sin{x}}=\frac{1}{\sin{x}}-1\]

Answer 2

\[\cos{x}=1-\sin{x} \\ \cos{x}+\sin{x}=1\\\cos{x}^2+2\cos{x}\sin{x} + \sin{x}^2=1\\2\cos{x}\sin{x}=0\\\sin{2x}=0\\2x=0,\pi,2\pi,3\pi, \dots ,k\pi\\x=0,0.5\pi.\pi,1.5\pi\]

Answer 3

Note that \[\cos{x}^2 +\sin{x}^2=1 \]

Answer 4

Note that since we have cotangent, sin(x) cannot be zero so this actually proves that there are no solutions in the interval.