15) csc theta=sec (4 theta+40^circ) [ theta=square^circ ]

1. Express sec in terms of sin <br /> $\sec x = \frac{1}{\cos x}$, $\csc \theta = \frac{1}{\sin \theta} \implies \frac{1}{\sin \theta} = \frac{1}{\cos(4\theta + 40^\circ)}$. <br /><br />2. Simplify equation <br /> $\sin \theta = \cos(4\theta + 40^\circ)$. <br /><br />3. Use complementary angle identity <br /> $\cos A = \sin(90^\circ - A) \implies \sin \theta = \sin(90^\circ - (4\theta + 40^\circ)) = \sin(50^\circ - 4\theta)$. <br /><br />4. Solve sine equality cases <br /> For $\sin X = \sin Y$: <br />1) $ \theta = 50^\circ - 4\theta + 360^\circ n \implies 5\theta = 50^\circ + 360^\circ n \implies \theta = 10^\circ + 72^\circ n$. <br />2) $\theta = 180^\circ - (50^\circ - 4\theta) + 360^\circ n \implies \theta = 130^\circ + 4\theta + 360^\circ n \implies -3\theta = 130^\circ + 360^\circ n \implies \theta = -\frac{130^\circ + 360^\circ n}{3}$. <br /><br />5. Find solutions in $[0^\circ, 360^\circ)$ <br /> From case 1: $n=0 \to 10^\circ$, $n=1 \to 82^\circ$, $n=2 \to 154^\circ$, $n=3 \to 226^\circ$, $n=4 \to 298^\circ$. <br />From case 2: <br />- $n=0$: $\theta = -43.\overline{3}^\circ$ (invalid), <br />- $n=1$: $\theta = 76.\overline{6}^\circ$, <br />- $n=2$: $\theta = 196.\overline{6}^\circ$, <br />- $n=3$: $\theta = 316.\overline{6}^\circ$.