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

Concept:
This is an Assertion-Reason question from Trigonometry. Instead of only substituting $\theta = 45^\circ$ and separately checking the Reason, let's first work out in general terms exactly when $\tan 2\theta$ fails to be defined, and then test with a fresh example whether the Reason is really the cause of that failure.

Step 1: Work out when tan $2\theta$ is undefined, in general.
By definition:
\[ \tan 2\theta = \frac{\sin 2\theta}{\cos 2\theta} \]
A fraction is undefined exactly when its denominator is zero. So $\tan 2\theta$ is undefined precisely when:
\[ \cos 2\theta = 0 \]

Step 2: Check whether this condition holds at $\theta = 45^\circ$.
At $\theta = 45^\circ$, we get $2\theta = 90^\circ$, and:
\[ \cos 90^\circ = 0 \]
So the condition $\cos 2\theta = 0$ is satisfied at $\theta = 45^\circ$, which means $\tan 2\theta$ is genuinely undefined there. Assertion (A) is true.

Step 3: Check the Reason on its own.
We know the standard values $\sin 90^\circ = 1$ and $\cos 90^\circ = 0$. Since $1 \neq 0$:
\[ \sin 90^\circ \neq \cos 90^\circ \]
This is a true statement, so Reason (R) is true on its own.

Step 4: Test if Reason (R) actually explains Assertion (A), using a fresh counterexample angle.
If "$\sin 90^\circ \neq \cos 90^\circ$" were truly the reason $\tan 90^\circ$ is undefined, then any angle where sine and cosine are unequal should also give an undefined tangent. Let's test this idea with a different angle, $60^\circ$.
We know $\sin 60^\circ = \frac{\sqrt{3}}{2}$ and $\cos 60^\circ = \frac{1}{2}$, and clearly $\sin 60^\circ \neq \cos 60^\circ$.
But $\tan 60^\circ = \sqrt{3}$, which is perfectly well defined.
So having unequal sine and cosine values does not, by itself, make the tangent undefined.

Step 5: Identify the real cause.
The true reason $\tan 90^\circ$ is undefined is that its denominator $\cos 90^\circ$ is exactly zero, not merely that sine and cosine differ in value at that angle. So Reason (R), although true, is not the correct explanation of Assertion (A).

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).