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: Look at how the tangent ratio behaves near 90 degrees.
In a right triangle, $\tan\theta = \frac{\text{side opposite }\theta}{\text{side adjacent }\theta}$. As the angle $\theta$ inside a right triangle grows toward $90^{\circ}$, the side adjacent to $\theta$ keeps shrinking toward zero while the opposite side stays roughly the same length.
Since the adjacent side is the denominator of the ratio, the ratio grows without any bound, so $\tan 90^{\circ}$ is not a fixed number. It is undefined.

Step 2: Apply this to the given Assertion.
We are told $\theta = 45^{\circ}$, so
\[ 2\theta = 2 \times 45^{\circ} = 90^{\circ} \]
By the reasoning in Step 1, $\tan 2\theta = \tan 90^{\circ}$ has no defined value.
So Assertion (A) is a true statement.

Step 3: Check the Reason using the unit circle.
Place a unit circle (radius $1$) with its centre at the origin. For any angle $\theta$ measured from the positive x axis, the point where the circle is cut gives $\cos\theta$ as its x coordinate and $\sin\theta$ as its y coordinate.
At $\theta = 90^{\circ}$, this point sits directly above the centre, on the positive y axis, at coordinates $(0, 1)$.
Reading off these coordinates: $\cos 90^{\circ} = 0$ (the x coordinate) and $\sin 90^{\circ} = 1$ (the y coordinate).
Since $1 \neq 0$, the statement $\sin 90^{\circ} \neq \cos 90^{\circ}$ is correct, so Reason (R) is also true.

Step 4: Decide if Reason (R) actually explains Assertion (A).
Assertion (A) is undefined specifically because the ratio $\frac{\sin 90^{\circ}}{\cos 90^{\circ}}$ has $0$ in the denominator, that is, because $\cos 90^{\circ} = 0$.
Reason (R) only tells us that $\sin 90^{\circ}$ and $\cos 90^{\circ}$ are two different numbers. It never points out that the cosine value itself is zero, which is the actual cause of the division problem.
So Reason (R), even though true, does not give the real explanation for why Assertion (A) holds.

Final Answer:
Both Assertion (A) and Reason (R) are true, but Reason (R) is not the correct explanation of Assertion (A). This matches option (B).