Problem Analysis

Given the trigonometric equation \(12 (\tan \theta + \cot \theta) = 25\) and the angle range \(45^\circ < \theta < 90^\circ\). The objective is to find the value of \((\cosec\theta + \sec\theta)\).

Step 1: Simplify the Equation

Rewrite the equation using \(\sin \theta\) and \(\cos \theta\):

\(\tan \theta + \cot \theta = \frac{\sin \theta}{\cos \theta} + \frac{\cos \theta}{\sin \theta} = \frac{\sin^2 \theta + \cos^2 \theta}{\sin \theta \cos \theta}\)

Using the fundamental trigonometric identity \(\sin^2 \theta + \cos^2 \theta = 1\):

\(\tan \theta + \cot \theta = \frac{1}{\sin \theta \cos \theta}\)

Substitute this back into the given equation:

\(12 \left( \frac{1}{\sin \theta \cos \theta} \right) = 25\)

Solve for the product \(\sin \theta \cos \theta\):

\(\sin \theta \cos \theta = \frac{12}{25}\)

Step 2: Find \(\sin \theta + \cos \theta\)

Consider the square of the sum \((\sin \theta + \cos \theta)\):

\((\sin \theta + \cos \theta)^2 = \sin^2 \theta + \cos^2 \theta + 2 \sin \theta \cos \theta\)

Substitute the known values \(\sin^2 \theta + \cos^2 \theta = 1\) and \(\sin \theta \cos \theta = \frac{12}{25}\):

\((\sin \theta + \cos \theta)^2 = 1 + 2 \left( \frac{12}{25} \right)\) \((\sin \theta + \cos \theta)^2 = 1 + \frac{24}{25}\) \((\sin \theta + \cos \theta)^2 = \frac{25 + 24}{25} = \frac{49}{25}\)

Given that \(45^\circ < \theta < 90^\circ\), both \(\sin \theta\) and \(\cos \theta\) are positive. Therefore, their sum is also positive:

\(\sin \theta + \cos \theta = \sqrt{\frac{49}{25}} = \frac{7}{5}\)

Step 3: Calculate \(\cosec\theta + \sec\theta\)

Express the target function \(\cosec\theta + \sec\theta\) in terms of \(\sin \theta\) and \(\cos \theta\):

\(\cosec\theta + \sec\theta = \frac{1}{\sin \theta} + \frac{1}{\cos \theta}\)

Combine the fractions:

\(\cosec\theta + \sec\theta = \frac{\cos \theta + \sin \theta}{\sin \theta \cos \theta}\)

Substitute the calculated values for \((\sin \theta + \cos \theta)\) and \((\sin \theta \cos \theta)\):

\(\cosec\theta + \sec\theta = \frac{7/5}{12/25}\)

Perform the division:

\(\cosec\theta + \sec\theta = \frac{7}{5} \times \frac{25}{12} = \frac{7 \times 5}{12} = \frac{35}{12}\)

Final Answer Determination

The derived value for \((\cosec\theta + \sec\theta)\) is \(\frac{35}{12}\).