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:

Always look to simplify expressions using identities like \(\sin^2 \theta + \cos^2 \theta = 1\) before calculating individual sine or cosine values. It usually saves a lot of arithmetic effort.

Answer 1

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

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

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

Thus,
\[ \sin\theta = \frac{8}{\sqrt{113}}, \quad \cos\theta = \frac{7}{\sqrt{113}} \]

Step 3: Evaluate the expression
\[ \frac{(1+\sin\theta)(1-\sin\theta)}{(1+\cos\theta)(1-\cos\theta)} \]

Note 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 cotθ
\[ \cot^2\theta = \left(\frac{7}{8}\right)^2 = \frac{49}{64} \]

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