\( \frac { 1 } { \sin \theta } + \frac { 1 } { \operatorname { L-c⿻ } \theta } \theta = \frac { 1 + \cos \theta } { \sin \theta } \)

Given the equation: \[ \frac{1}{\sin \theta} + \frac{1}{\operatorname{L-c⿻} \theta} \theta = \frac{1 + \cos \theta}{\sin \theta} \] We can simplify the equation by combining the terms on the left side: \[ \frac{1}{\sin \theta} + \frac{\theta}{\operatorname{L-c⿻} \theta} = \frac{1 + \cos \theta}{\sin \theta} \] Now, we can solve this equation for \( \theta \). Solve the equation by following steps: - step0: Solve for \(\theta\): \(\frac{1}{\sin\left(\theta \right)}+\frac{\theta }{\theta }=\frac{1+\cos\left(\theta \right)}{\sin\left(\theta \right)}\) - step1: Find the domain: \(\frac{1}{\sin\left(\theta \right)}+\frac{\theta }{\theta }=\frac{1+\cos\left(\theta \right)}{\sin\left(\theta \right)},\theta \in \left(-\infty,0\right)\cap \theta \neq k\pi ,k \in \mathbb{Z}\cup \theta \in \left(0,+\infty\right)\cap \theta \neq k\pi ,k \in \mathbb{Z}\) - step2: Divide the terms: \(\frac{1}{\sin\left(\theta \right)}+1=\frac{1+\cos\left(\theta \right)}{\sin\left(\theta \right)}\) - step3: Multiply both sides of the equation by LCD: \(\left(\frac{1}{\sin\left(\theta \right)}+1\right)\sin\left(\theta \right)=\frac{1+\cos\left(\theta \right)}{\sin\left(\theta \right)}\times \sin\left(\theta \right)\) - step4: Simplify the equation: \(1+\sin\left(\theta \right)=1+\cos\left(\theta \right)\) - step5: Move the expression to the left side: \(1+\sin\left(\theta \right)-1=\cos\left(\theta \right)\) - step6: Calculate: \(\sin\left(\theta \right)=\cos\left(\theta \right)\) - step7: Divide both sides: \(\frac{\sin\left(\theta \right)}{\cos\left(\theta \right)}=1\) - step8: Divide the terms: \(\tan\left(\theta \right)=1\) - step9: Use the inverse trigonometric function: \(\theta =\arctan\left(1\right)\) - step10: Calculate: \(\theta =\frac{\pi }{4}\) - step11: Add the period: \(\theta =\frac{\pi }{4}+k\pi ,k \in \mathbb{Z}\) - step12: Check if the solution is in the defined range: \(\theta =\frac{\pi }{4}+k\pi ,k \in \mathbb{Z},\theta \in \left(-\infty,0\right)\cap \theta \neq k\pi ,k \in \mathbb{Z}\cup \theta \in \left(0,+\infty\right)\cap \theta \neq k\pi ,k \in \mathbb{Z}\) - step13: Find the intersection: \(\theta =\frac{\pi }{4}+k\pi ,k \in \left(-\frac{1}{4},+\infty\right),k \in \mathbb{Z}\cup k \in \left(-\infty,-\frac{1}{4}\right),k \in \mathbb{Z}\) The solution to the equation is \( \theta = \frac{\pi}{4} + k\pi \), where \( k \) is an integer. This means that the value of \( \theta \) can be any multiple of \( \frac{\pi}{4} \) plus an integer multiple of \( \pi \).