Question

If \(\cot \theta = \frac{7}{8}\), then evaluate \[ \frac{(1 + \sin \theta)(1 - \sin \theta)}{(1 + \cos \theta)(1 - \cos \theta)} \]

Hint:

For trigonometric identities, express \(\sin\) and \(\cos\) in terms of \(\cot\) and simplify using the Pythagorean identity \(\sin^2 \theta + \cos^2 \theta = 1\).

Answer 1

Step 1: Express \(\sin \theta\) and \(\cos \theta\) using \(\cot \theta\).
We are given \(\cot \theta = \frac{7}{8}\). We know that: \[ \cot \theta = \frac{\cos \theta}{\sin \theta} \] Thus, \(\cos \theta = 7k\) and \(\sin \theta = 8k\) for some constant \(k\). Also, since \(\sin^2 \theta + \cos^2 \theta = 1\), we substitute: \[ (8k)^2 + (7k)^2 = 1 \quad \Rightarrow \quad 64k^2 + 49k^2 = 1 \quad \Rightarrow \quad 113k^2 = 1 \quad \Rightarrow \quad k = \frac{1}{\sqrt{113}} \]
Step 2: Substitute values of \(\sin \theta\) and \(\cos \theta\) into the expression.
Now that we know \(k = \frac{1}{\sqrt{113}}\), we substitute \(\sin \theta = \frac{8}{\sqrt{113}}\) and \(\cos \theta = \frac{7}{\sqrt{113}}\) into the expression: \[ \frac{(1 + \sin \theta)(1 - \sin \theta)}{(1 + \cos \theta)(1 - \cos \theta)} = \frac{(1 + \frac{8}{\sqrt{113}})(1 - \frac{8}{\sqrt{113}})}{(1 + \frac{7}{\sqrt{113}})(1 - \frac{7}{\sqrt{113}})} \]
Step 3: Simplify the expression.
Use the difference of squares formula to simplify both the numerator and the denominator: \[ = \frac{1 - \left(\frac{8}{\sqrt{113}}\right)^2}{1 - \left(\frac{7}{\sqrt{113}}\right)^2} = \frac{1 - \frac{64}{113}}{1 - \frac{49}{113}} = \frac{\frac{113 - 64}{113}}{\frac{113 - 49}{113}} = \frac{49}{64} \] Thus, the value of the expression is: \[ \frac{49}{64} \]