Question

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:

An expression of the form $\frac{f(x)}{g(x)}$ becomes undefined specifically when the denominator $g(x) = 0$.
Always look for the condition "denominator $= 0$" as the correct explanation for undefined trigonometric terms (like $\tan \theta$ or $\sec \theta$).

Answer 1

The Correct Option is B

Step 1: Understanding the Question:
This is an "Assertion-Reason" type question from the topic "Trigonometry".
We need to determine the truth values of the Assertion (A) and the Reason (R) and see if there is a direct causal relationship between them.

Step 2: Key Formula or Approach:
Let's analyze both statements using standard trigonometric properties:

• Assertion (A): "$\tan 2\theta$ is not defined at $\theta = 45^\circ$."
Substitute $\theta = 45^\circ$ into the expression:
\[ \tan(2 \times 45^\circ) = \tan(90^\circ) \] Since $\tan(90^\circ)$ is undefined, Assertion (A) is True.

• Reason (R): "$\sin 90^\circ \neq \cos 90^\circ$."
We know that $\sin 90^\circ = 1$ and $\cos 90^\circ = 0$. Since $1 \neq 0$, the statement is True.

Step 3: Detailed Explanation:

• Let's evaluate why $\tan 90^\circ$ is not defined. By definition:
\[ \tan 90^\circ = \frac{\sin 90^\circ}{\cos 90^\circ} \] Since $\cos 90^\circ = 0$, the expression involves division by zero, which is mathematically undefined.

• The fact that $\sin 90^\circ$ is not equal to $\cos 90^\circ$ is a true mathematical statement.

• However, inequality between sine and cosine does not cause a tangent function to be undefined. For example, $\sin 30^\circ \neq \cos 30^\circ$, but $\tan 30^\circ = \frac{1}{\sqrt{3}}$, which is perfectly defined.

• The real reason $\tan 90^\circ$ is undefined is specifically because the denominator $\cos 90^\circ = 0$.

• Therefore, while both Assertion (A) and Reason (R) are true statements, the Reason is not the correct explanation for the Assertion.

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