Question

If \(\frac{1 + \cos \theta + \sin \theta}{2 \sin \theta} = \frac{\lambda}{1 - \cos \theta + \sin \theta}\), then \(\lambda - 1 =\)

• A. \(\sin\theta\)
• B. \(\cos\theta\)
• C. \(\tan\theta\)
• D. \(\cot\theta\)

Hint:

Whenever expressions contain \(1\pm \sin\theta\) and \(1\pm \cos\theta\), try expansion using identities before substitution.

Answer 1

The Correct Option is C

Step 1: Cross multiply. \[ (1+\cos\theta+\sin\theta)(1-\cos\theta+\sin\theta)=2\lambda \sin\theta \]

Step 2: Expand LHS. \[ (1+\sin\theta)^2-\cos^2\theta \] \[ =1+2\sin\theta+\sin^2\theta-\cos^2\theta \] Using identity: \[ \sin^2\theta-\cos^2\theta=-\cos2\theta \] After simplification: \[ 2\lambda\sin\theta = 2\sin\theta(1+\tan\theta) \] \[ \lambda = 1+\tan\theta \] \[ \lambda-1=\tan\theta \] \[ \boxed{\tan\theta} \]