Simplify tanθ and cotθ Equation

Given the equation \(12 (\tan \theta + \cot \theta) = 25\) and the range \(45^\circ < \theta < 90^\circ\). We need to find the value of \((\sin \theta - \cos \theta)\).

Rewrite \(\tan \theta\) and \(\cot \theta\) using \(\sin \theta\) and \(\cos \theta\):

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

Substitute into the equation:

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

Find a common denominator:

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

Apply the identity \(\sin^2 \theta + \cos^2 \theta = 1\):

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

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

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

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

Calculate (sinθ - cosθ)² Value

Consider the square of the target expression \((\sin \theta - \cos \theta)\):

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

Group terms and use the identity \(\sin^2 \theta + \cos^2 \theta = 1\):

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

\((\sin \theta - \cos \theta)^2 = 1 - 2 \sin \theta \cos \theta\)

Substitute the value of \(\sin \theta \cos \theta\):

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

\((\sin \theta - \cos \theta)^2 = \frac{25 - 24}{25} = \frac{1}{25}\)

Determine (sinθ - cosθ) Sign

Take the square root of both sides:

\(\sin \theta - \cos \theta = \pm \sqrt{\frac{1}{25}} = \pm \frac{1}{5}\)

Analyze the sign based on the given range \(45^\circ < \theta < 90^\circ\):

• In this range, \(\theta\) is in the first quadrant.
• \(\sin \theta\) and \(\cos \theta\) are both positive.
• For angles greater than \(45^\circ\) (up to \(90^\circ\)), \(\sin \theta > \cos \theta\).

Therefore, \((\sin \theta - \cos \theta)\) must be positive.

Final (sinθ - cosθ) Result

Select the positive value:

\(\sin \theta - \cos \theta = \frac{1}{5}\)