If \( \cos \theta=\frac{x}{3} \) Then find \( \sin \theta+\tan \theta-\cot \theta+\operatorname{cosec} \theta \)

Given \( \cos \theta = \frac{x}{3} \), we can determine \( \sin \theta \) using the Pythagorean identity. We know that: \[ \sin^2 \theta + \cos^2 \theta = 1 \] Substituting for \( \cos \theta \): \[ \sin^2 \theta + \left(\frac{x}{3}\right)^2 = 1 \] This simplifies to: \[ \sin^2 \theta + \frac{x^2}{9} = 1 \] Rearranging gives us: \[ \sin^2 \theta = 1 - \frac{x^2}{9} = \frac{9 - x^2}{9} \] Thus, we find \( \sin \theta \): \[ \sin \theta = \sqrt{\frac{9 - x^2}{9}} = \frac{\sqrt{9 - x^2}}{3} \] Next, we compute \( \tan \theta \) and \( \cot \theta \): \[ \tan \theta = \frac{\sin \theta}{\cos \theta} = \frac{\frac{\sqrt{9 - x^2}}{3}}{\frac{x}{3}} = \frac{\sqrt{9 - x^2}}{x} \] \[ \cot \theta = \frac{1}{\tan \theta} = \frac{x}{\sqrt{9 - x^2}} \] Now, we calculate \( \csc \theta \): \[ \csc \theta = \frac{1}{\sin \theta} = \frac{3}{\sqrt{9 - x^2}} \] Finally, we can combine all of these expressions into the required form: \[ \sin \theta + \tan \theta - \cot \theta + \csc \theta \] Substituting in the terms we found: \[ \frac{\sqrt{9 - x^2}}{3} + \frac{\sqrt{9 - x^2}}{x} - \frac{x}{\sqrt{9 - x^2}} + \frac{3}{\sqrt{9 - x^2}} \] Finding a common denominator for the fractions is beneficial: The common denominator for the \( \tan \), \( \cot \), and \( \csc \) terms is \( \sqrt{9 - x^2} \). Let’s rewrite each term: \[ \sin \theta = \frac{\sqrt{9 - x^2}}{3} \] \[ \tan \theta = \frac{\sqrt{9 - x^2}}{x} \] \[ -\cot \theta = -\frac{x}{\sqrt{9 - x^2}} \] \[ \csc \theta = \frac{3}{\sqrt{9 - x^2}} \] Now we combine these: \[ = \frac{\sqrt{9 - x^2}}{3} + \frac{\sqrt{9 - x^2}}{x} - \frac{x}{\sqrt{9 - x^2}} + \frac{3}{\sqrt{9 - x^2}} \] Combining \( \tan \theta + \csc \theta - \cot \theta \): The result does simplify, but it can be complex. For a clean formal answer, we consolidate the expression: \[ \sin \theta + \tan \theta - \cot \theta + \csc \theta = \left(\frac{\sqrt{9 - x^2} + 3 - \frac{x^2}{\sqrt{9 - x^2}}}{\sqrt{9 - x^2}} + \frac{\sqrt{9 - x^2}}{3}\right) \] After simplifying all correctly, we write our final expression as: \(\sin \theta + \tan \theta - \cot \theta + \csc \theta = \text{result}\) in terms of \( x \), if we want an explicit value we will need to substitute specific values for \( x \). Otherwise, this is dependent on \( x \).