Solving Sec θ - Cosec θ = 0

We are given the equation $\sec \theta - \text{cosec} \theta = 0$.

• Rewrite the equation as $\sec \theta = \text{cosec} \theta$.
• Express in terms of $\sin \theta$ and $\cos \theta$: $\frac{1}{\cos \theta} = \frac{1}{\sin \theta}$.
• This implies $\sin \theta = \cos \theta$.
• Dividing both sides by $\cos \theta$ (assuming $\cos \theta \neq 0$), we get $\frac{\sin \theta}{\cos \theta} = 1$.
• Therefore, $\tan \theta = 1$.
• The principal value for $\theta$ that satisfies $\tan \theta = 1$ is $\theta = 45^\circ$ or $\frac{\pi}{4}$ radians.

Evaluating the Expression

We need to find the value of $2\sin 2\theta - 3\cos 2\theta$. Substitute $\theta = 45^\circ$ into the expression.

• First, calculate $2\theta$: $2\theta = 2 \times 45^\circ = 90^\circ$.
• Now calculate $\sin 2\theta$ and $\cos 2\theta$:
  • $\sin 2\theta = \sin 90^\circ = 1$
  • $\cos 2\theta = \cos 90^\circ = 0$
• Substitute these values back into the expression: $2\sin 2\theta - 3\cos 2\theta = 2(1) - 3(0)$
• Simplify the expression: $2 - 0 = 2$.

The value of the expression is $2$.