Question

If \(\cot \theta = \frac{7}{8}\), then find the value of \(\frac{(1+\sin \theta)(1-\sin \theta)}{(1+\cos \theta)(1-\cos \theta)}\).

Hint:

For part (B), don't waste time finding \(\sin\) and \(\cos\) individually; always simplify the algebraic expression using trigonometric identities first!

Answer 1

Given:
\(\cot \theta = \frac{7}{8}\)

Step 1: Convert cot to sin and cos
\[ \cot\theta = \frac{\cos\theta}{\sin\theta} = \frac{7}{8} \]
Let: \(\cos\theta = 7k\), \(\sin\theta = 8k\)

Step 2: Use identity
\(\sin^2\theta + \cos^2\theta = 1\)
Substitute values:
\[ (8k)^2 + (7k)^2 = 1 \] \[ 64k^2 + 49k^2 = 1 \] \[ 113k^2 = 1 \] \[ k = \frac{1}{\sqrt{113}} \]

Thus,
\(\sin\theta = \frac{8}{\sqrt{113}}\)
\(\cos\theta = \frac{7}{\sqrt{113}}\)

Step 3: Simplify the expression
\[ \frac{(1+\sin\theta)(1-\sin\theta)}{(1+\cos\theta)(1-\cos\theta)} \]
Use identities:
\((1+\sin\theta)(1-\sin\theta) = 1 - \sin^2\theta = \cos^2\theta\)
\((1+\cos\theta)(1-\cos\theta) = 1 - \cos^2\theta = \sin^2\theta\)

So the expression becomes:
\[ \frac{\cos^2\theta}{\sin^2\theta} = \cot^2\theta \]
Step 4: Substitute the value of cot
\[ \cot^2\theta = \left(\frac{7}{8}\right)^2 = \frac{49}{64} \]

Final Answer:
\[ \boxed{\frac{49}{64}} \]