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

Step 1: Solving OR (B):
1. Simplify the expression using \((a+b)(a-b) = a^2 - b^2\): \[ \frac{1 - \sin^2 \theta}{1 - \cos^2 \theta} \] 2. Use identity \(\sin^2 \theta + \cos^2 \theta = 1\): \[ \frac{\cos^2 \theta}{\sin^2 \theta} = \cot^2 \theta \] 3. Substitute \(\cot \theta = 7/8\): \[ (7/8)^2 = 49/64 \]
Step 2: Final Answer (OR):
The value is \(\frac{49}{64}\).