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

Step 1: Understanding the Concept:
The given expression can be simplified using basic trigonometric identities before substituting the values.
Step 2: Key Formula or Approach:
Algebraic identity: \((a+b)(a-b) = a^2 - b^2\)
Trigonometric identities: \(1 - \sin^2 \theta = \cos^2 \theta\) and \(1 - \cos^2 \theta = \sin^2 \theta\)
Relationship: \(\cot \theta = \frac{\cos \theta}{\sin \theta}\)
Step 3: Detailed Explanation:
The expression is:
\[ E = \frac{(1+\sin \theta)(1-\sin \theta)}{(1+\cos \theta)(1-\cos \theta)} \]
Using \((a+b)(a-b) = a^2 - b^2\):
\[ E = \frac{1 - \sin^2 \theta}{1 - \cos^2 \theta} \]
Applying trigonometric identities:
\[ E = \frac{\cos^2 \theta}{\sin^2 \theta} = \left(\frac{\cos \theta}{\sin \theta}\right)^2 = \cot^2 \theta \]
Given \(\cot \theta = \frac{7}{8}\).
\[ E = \left(\frac{7}{8}\right)^2 = \frac{49}{64} \]
Step 4: Final Answer:
The value of the expression is \(\frac{49}{64}\).