Q3(ii):

Choose the correct option. Justify your choice. (ii) $(1 + \tan \theta + \sec \theta) (1 + \cot \theta – \text{cosec } \theta) =$

Given: The trigonometric expression $(1 + \tan \theta + \sec \theta)(1 + \cot \theta - \text{cosec } \theta)$.

To Find: The simplified value of the given expression.

Step 1: Expressing all trigonometric ratios in terms of sine and cosine.

We use the following fundamental trigonometric identities:

$\tan \theta = \frac{\sin \theta}{\cos \theta}$

$\sec \theta = \frac{1}{\cos \theta}$

$\cot \theta = \frac{\cos \theta}{\sin \theta}$

$\text{cosec } \theta = \frac{1}{\sin \theta}$

Substituting these into the expression:

Expression $= \left(1 + \frac{\sin \theta}{\cos \theta} + \frac{1}{\cos \theta}\right) \left(1 + \frac{\cos \theta}{\sin \theta} - \frac{1}{\sin \theta}\right)$

Step 2: Simplifying the terms inside each bracket.

For the first bracket, find a common denominator ($\cos \theta$):

$\left(\frac{\cos \theta + \sin \theta + 1}{\cos \theta}\right)$

For the second bracket, find a common denominator ($\sin \theta$):

$\left(\frac{\sin \theta + \cos \theta - 1}{\sin \theta}\right)$

Step 3: Multiplying the two fractions.

Expression $= \frac{(\sin \theta + \cos \theta + 1)(\sin \theta + \cos \theta - 1)}{\sin \theta \cos \theta}$

Step 4: Applying the algebraic identity $(a + b)(a - b) = a^2 - b^2$.

Let $a = (\sin \theta + \cos \theta)$ and $b = 1$.

Numerator $= (\sin \theta + \cos \theta)^2 - (1)^2$

Expanding $(\sin \theta + \cos \theta)^2$ using $(a + b)^2 = a^2 + b^2 + 2ab$:

Numerator $= (\sin^2 \theta + \cos^2 \theta + 2 \sin \theta \cos \theta) - 1$

Step 5: Using the Pythagorean identity $\sin^2 \theta + \cos^2 \theta = 1$.

Numerator $= (1 + 2 \sin \theta \cos \theta) - 1$

Numerator $= 2 \sin \theta \cos \theta$

Step 6: Final simplification.

Expression $= \frac{2 \sin \theta \cos \theta}{\sin \theta \cos \theta}$

Canceling the common terms $\sin \theta \cos \theta$ (assuming $\sin \theta \neq 0$ and $\cos \theta \neq 0$):

Expression $= 2$

Final Answer: 2