Solving a Trigonometric Identity to Find General Angle Theta
<p>Given: \( \cot \theta + \tan \theta = 2 \csc \theta \)</p>
<p>Multiply each side by \( \sin \theta \) to simplify:</p>
<p>\( \sin \theta \cot \theta + \sin \theta \tan \theta = 2 \)</p>
<p>Using the identities \( \cot \theta = \frac{1}{\tan \theta} \) and \( \tan \theta = \frac{\sin \theta}{\cos \theta} \), we get:</p>
<p>\( \frac{\sin \theta}{\tan \theta} + \sin \theta \cdot \frac{\sin \theta}{\cos \theta} = 2 \)</p>
<p>\( \frac{\sin^2 \theta + \sin^2 \theta \tan^2 \theta}{\sin \theta \tan \theta} = 2 \)</p>
<p>Using the identity \( \sin^2 \theta + \tan^2 \theta = \frac{1}{\cos^2 \theta} \), we further simplify:</p>
<p>\( \frac{1 + \tan^2 \theta}{\tan \theta} \cdot \sin \theta = 2 \)</p>
<p>Applying the Pythagorean identity \( 1 + \tan^2 \theta = \sec^2 \theta \), we have:</p>
<p>\( \frac{\sec^2 \theta}{\tan \theta} \cdot \sin \theta = 2 \)</p>
<p>Since \( \sec^2 \theta = \frac{1}{\cos^2 \theta} \) and \( \tan \theta = \frac{\sin \theta}{\cos \theta} \), we get:</p>
<p>\( \frac{\sin \theta}{\cos^3 \theta} = 2 \)</p>
<p>Now, let's isolate \( \cos \theta \):</p>
<p>\( \sin \theta = 2 \cos^3 \theta \)</p>
<p>\( \frac{\sin \theta}{2} = \cos^3 \theta \)</p>
<p>Take the cube root:</p>
<p>\( \sqrt[3]{\frac{\sin \theta}{2}} = \cos \theta \)</p>
<p>Note that finding a general solution for \( \theta \) from this equation is not straightforward and involves iterative or numerical methods. Since we are only required to express a relation in terms of \( \theta \), we leave the equation as the final form, which expresses \( \cos \theta \) in terms of \( \sin \theta \):</p>
<p>\( \cos \theta = \sqrt[3]{\frac{\sin \theta}{2}} \)</p>
<p>To find the general solutions for \( \theta \), one would typically look at where \( \sin \theta \) and \( \cos \theta \) intersect at the values provided by the equation within the domain \( [0, 2\pi) \) or \( [0^\circ, 360^\circ) \). Since the original equation does not give us an explicit general solution, we maintain the final step as converting the relation between \( \sin \theta \) and \( \cos \theta \).</p>
