If  \( \tan \theta = \frac{7}{8} \), then evaluate \(\frac{(1 + \sin \theta)(1 - \sin \theta)}{(1 + \cos \theta)(1 - \cos \theta)(\cot \theta)}\)

Given:

\(\tan\theta = \frac{7}{8}\)

Expression to evaluate: \(\dfrac{(1 + \sin\theta)(1 - \sin\theta)}{(1 + \cos\theta)(1 - \cos\theta)(\cot\theta)}\)

Formula used:

1. (a + b)(a - b) = a² - b²

2. \(\sin^2\theta + \cos^2\theta = 1\)

⇒ \(1 - \sin^2\theta = \cos^2\theta\)

⇒ \(1 - \cos^2\theta = \sin^2\theta\)

3. \(\cot\theta = \dfrac{\cos\theta}{\sin\theta}\)

4. \(\cot\theta = \dfrac{1}{\tan\theta}\)

Calculations:

Simplify the numerator:

Numerator = \((1 + \sin\theta)(1 - \sin\theta)\)

⇒ Numerator = \(1^2 - \sin^2\theta\)

⇒ Numerator = \(1 - \sin^2\theta\)

⇒ Numerator = \(\cos^2\theta\)

Simplify the denominator:

Denominator = \((1 + \cos\theta)(1 - \cos\theta)(\cot\theta)\)

⇒ Denominator = \((1^2 - \cos^2\theta)(\cot\theta)\)

⇒ Denominator = \((1 - \cos^2\theta)(\cot\theta)\)

⇒ Denominator = \(\sin^2\theta \times \cot\theta\)

Substitute \(\cot\theta = \dfrac{\cos\theta}{\sin\theta}\):

⇒ Denominator = \(\sin^2\theta \times \dfrac{\cos\theta}{\sin\theta}\)

⇒ Denominator = \(\sin\theta \cos\theta\)

Now, substitute the simplified numerator and denominator into the expression:

Expression = \(\dfrac{\cos^2\theta}{\sin\theta \cos\theta}\)

Cancel out  \(\cos\theta\) from numerator and denominator:

⇒ Expression = \(\dfrac{\cos\theta}{\sin\theta}\)

⇒ Expression = \(\cot\theta\)

Given \(\tan\theta = \dfrac{7}{8}\).

Since \(\cot\theta = \dfrac{1}{\tan\theta}\):

⇒ Expression = \(\dfrac{1}{\frac{7}{8}}\)

⇒ Expression = \(\dfrac{8}{7}\)

∴ The value of the expression is \(\dfrac{8}{7}\).