Let's solve the given problem step by step.
Given:
[tex]\[ 21 \operatorname{cosec} \theta = 29 \][/tex]
### Step 1: Find [tex]\(\sin \theta\)[/tex]
Recall that [tex]\(\operatorname{cosec} \theta = \frac{1}{\sin \theta}\)[/tex], so we can rewrite the given equation as:
[tex]\[ 21 \cdot \frac{1}{\sin \theta} = 29 \][/tex]
Solving for [tex]\(\sin \theta\)[/tex]:
[tex]\[ \sin \theta = \frac{21}{29} \][/tex]
### Step 2: Find [tex]\(\cos \theta\)[/tex]
We use the Pythagorean identity:
[tex]\[ \sin^2 \theta + \cos^2 \theta = 1 \][/tex]
Substitute [tex]\(\sin \theta = \frac{21}{29}\)[/tex]:
[tex]\[ \left( \frac{21}{29} \right)^2 + \cos^2 \theta = 1 \][/tex]
[tex]\[ \frac{441}{841} + \cos^2 \theta = 1 \][/tex]
Solving for [tex]\(\cos^2 \theta\)[/tex]:
[tex]\[ \cos^2 \theta = 1 - \frac{441}{841} \][/tex]
[tex]\[ \cos^2 \theta = \frac{841}{841} - \frac{441}{841} = \frac{400}{841} \][/tex]
Thus:
[tex]\[ \cos \theta = \sqrt{ \frac{400}{841}} = \frac{20}{29} \][/tex]
Now we have:
[tex]\[ \sin \theta = \frac{21}{29} \][/tex]
[tex]\[ \cos \theta = \frac{20}{29} \][/tex]
### Step 3: Calculate expressions
#### (i) [tex]\(\frac{\cos^2 \theta - \sin^2 \theta}{1 - 2 \sin^2 \theta}\)[/tex]
First, compute [tex]\(\cos^2 \theta - \sin^2 \theta\)[/tex]:
[tex]\[ \cos^2 \theta - \sin^2 \theta = \left( \frac{20}{29} \right)^2 - \left( \frac{21}{29} \right)^2 \][/tex]
[tex]\[ \cos^2 \theta - \sin^2 \theta = \frac{400}{841} - \frac{441}{841} = \frac{400 - 441}{841} = \frac{-41}{841} \][/tex]
Then, compute [tex]\(1 - 2 \sin^2 \theta\)[/tex]:
[tex]\[ 1 - 2 \sin^2 \theta = 1 - 2 \left( \frac{21}{29} \right)^2 \][/tex]
[tex]\[ 1 - 2 \sin^2 \theta = 1 - 2 \cdot \frac{441}{841} \][/tex]
[tex]\[ 1 - 2 \sin^2 \theta = 1 - \frac{882}{841} = \frac{841}{841} - \frac{882}{841} = \frac{-41}{841} \][/tex]
Now the expression:
[tex]\[ \frac{\cos^2 \theta - \sin^2 \theta}{1 - 2 \sin^2 \theta} = \frac{\frac{-41}{841}}{\frac{-41}{841}} = 1 \][/tex]
So the value is:
[tex]\[ \boxed{1} \][/tex]
#### (ii) [tex]\(\frac{2 \cos^2 \theta - 1}{\cos^2 \theta - \sin^2 \theta}\)[/tex]
First, compute [tex]\(2 \cos^2 \theta - 1\)[/tex]:
[tex]\[ 2 \cos^2 \theta - 1 = 2 \left( \frac{20}{29} \right)^2 - 1 \][/tex]
[tex]\[ 2 \cos^2 \theta - 1 = 2 \cdot \frac{400}{841} - 1 = \frac{800}{841} - \frac{841}{841} = \frac{800-841}{841} = \frac{-41}{841} \][/tex]
Now we already have [tex]\(\cos^2 \theta - \sin^2 \theta\)[/tex] from part (i):
[tex]\[ \cos^2 \theta - \sin^2 \theta = \frac{-41}{841} \][/tex]
So the expression:
[tex]\[ \frac{2 \cos^2 \theta - 1}{\cos^2 \theta - \sin^2 \theta} = \frac{\frac{-41}{841}}{\frac{-41}{841}} = 1 \][/tex]
So the value is:
[tex]\[ \boxed{1} \][/tex]
Therefore, the values are:
(i) [tex]\(\frac{\cos^2 \theta - \sin^2 \theta}{1 - 2 \sin^2 \theta} = \boxed{1}\)[/tex]
(ii) [tex]\(\frac{2 \cos^2 \theta - 1}{\cos^2 \theta - \sin^2 \theta} = \boxed{1}\)[/tex]
