Question

Questions number 19 and 20 are Assertion and Reason based questions. Two statements are given, one labelled as Assertion (A) and the other is labelled as Reason (R). Select the correct answer to these questions from the codes (A), (B), (C) and (D) as given below.
Assertion (A) : tan 2\(\theta\) is not defined at \(\theta\) = 45\(^{\circ}\).
Reason (R) : sin 90\(^{\circ}\) \(\neq\) cos 90\(^{\circ}\).

• A. Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of the Assertion (A).
• B. Both Assertion (A) and Reason (R) are true, but Reason (R) is not the correct explanation of the Assertion (A).
• C. Assertion (A) is true, but Reason (R) is false.
• D. Assertion (A) is false, but Reason (R) is true.

Hint:

In Assertion-Reason questions, read both statements as separate statements first.
If both are true, insert the word "because" between them:
"tan 2\(\theta\) is not defined at \(\theta = 45^{\circ}\) because \(\sin 90^{\circ} \neq \cos 90^{\circ}\)."
This immediately helps you see that while both parts are true facts, the second part does not explain the first part. This confirms option (B).

Answer 1

The Correct Option is B

Step 1: Understanding the Question:
The topic of this question is Trigonometry (Trigonometric functions and Assertion-Reason analysis).
We are given an Assertion statement about the undefined nature of \( \tan 2\theta \) at \( \theta = 45^{\circ} \) and a Reason statement comparing the values of \( \sin 90^{\circ} \) and \( \cos 90^{\circ} \).
We need to evaluate the truth value of both statements independently and then determine if the Reason is the correct logical explanation for the Assertion.

Step 2: Key Formula or Approach:
- Recall the trigonometric identity:
\[ \tan x = \frac{\sin x}{\cos x} \]
- A fraction of the form \( \frac{p}{q} \) is not defined when the denominator is equal to zero, i.e., \( q = 0 \).
- Evaluate the exact values of \( \sin 90^{\circ} \) and \( \cos 90^{\circ} \).

Step 3: Detailed Explanation:
1. Evaluate Assertion (A):
We are given \( \theta = 45^{\circ} \). Let us compute the value of the angle \( 2\theta \):
\[ 2\theta = 2 \times 45^{\circ} = 90^{\circ} \]
Now, evaluate the function value:
\[ \tan 2\theta = \tan 90^{\circ} \]
Since the tangent of \( 90^{\circ} \) is undefined (it approaches infinity as the angle approaches \( 90^{\circ} \)), the statement "tan 2\(\theta\) is not defined at \(\theta\) = 45\(^{\circ}\)" is True.
2. Evaluate Reason (R):
Let us find the values of \( \sin 90^{\circ} \) and \( \cos 90^{\circ} \):
- From standard values, \( \sin 90^{\circ} = 1 \).
- From standard values, \( \cos 90^{\circ} = 0 \).
Since \( 1 \neq 0 \), the statement \( \sin 90^{\circ} \neq \cos 90^{\circ} \) is mathematically correct and therefore True.
3. Check for Logical Explanation:
Why is \( \tan 90^{\circ} \) not defined?
Mathematically:
\[ \tan 90^{\circ} = \frac{\sin 90^{\circ}}{\cos 90^{\circ}} = \frac{1}{0} \]
Division by zero is undefined, which is why \( \tan 90^{\circ} \) is not defined.
The direct reason for it being undefined is that \( \cos 90^{\circ} = 0 \).
The fact that \( \sin 90^{\circ} \neq \cos 90^{\circ} \) is a true statement, but it does not explain why the division by zero occurs or why the tangent function becomes undefined.
Therefore, Reason (R) is not the correct explanation of Assertion (A).

Step 4: Final Answer:
Both Assertion (A) and Reason (R) are true, but Reason (R) is not the correct explanation of Assertion (A), which corresponds to option (B).