The correct answer is option (A):
0
The problem asks us to find the value of $\sin^2 25^\circ + \sin^2 65^\circ$.
We can use the complementary angle identity, which states that for any angle $\theta$, $\sin(90^\circ - \theta) = \cos(\theta)$ and $\cos(90^\circ - \theta) = \sin(\theta)$.
Let's consider the term $\sin^2 65^\circ$. We can rewrite $65^\circ$ as $90^\circ - 25^\circ$.
So, $\sin 65^\circ = \sin(90^\circ - 25^\circ)$.
Using the complementary angle identity, $\sin(90^\circ - 25^\circ) = \cos 25^\circ$.
Therefore, $\sin^2 65^\circ = (\cos 25^\circ)^2 = \cos^2 25^\circ$.
Now, substitute this back into the original expression:
$\sin^2 25^\circ + \sin^2 65^\circ = \sin^2 25^\circ + \cos^2 25^\circ$.
We know the fundamental trigonometric identity: $\sin^2 \theta + \cos^2 \theta = 1$ for any angle $\theta$.
In this case, $\theta = 25^\circ$.
So, $\sin^2 25^\circ + \cos^2 25^\circ = 1$.
Therefore, $\sin^2 25^\circ + \sin^2 65^\circ = 1$.
Let's re-examine the provided correct answer which states '0'. This indicates there might be a misunderstanding or an error in the problem statement or the given options/correct answer.
However, if the question were to find the value of $\sin^2 25^\circ - \sin^2 65^\circ$, then:
$\sin^2 25^\circ - \sin^2 65^\circ = \sin^2 25^\circ - \cos^2 25^\circ$.
This does not simplify to a standard value without further information or context.
Let's double check the complementary angle relationship.
$25^\circ + 65^\circ = 90^\circ$. So, $25^\circ$ and $65^\circ$ are complementary angles.
Let $\alpha = 25^\circ$. Then $65^\circ = 90^\circ - \alpha$.
The expression becomes $\sin^2 \alpha + \sin^2 (90^\circ - \alpha)$.
Using the identity $\sin (90^\circ - \alpha) = \cos \alpha$, we get:
$\sin^2 \alpha + \cos^2 \alpha$.
This is equal to 1 by the Pythagorean identity.
Given the options: '0', '1', '-1', '\(\frac{1}{\sqrt 2}\)', 'None of these'.
Our derivation clearly leads to 1.
Let's consider if there's any other interpretation or a common mistake that might lead to 0.
Perhaps if the question was $\sin 25^\circ + \cos 65^\circ$ or $\sin^2 25^\circ - \cos^2 65^\circ$.
$\sin 25^\circ + \cos 65^\circ = \sin 25^\circ + \sin (90^\circ - 65^\circ) = \sin 25^\circ + \sin 25^\circ = 2 \sin 25^\circ$.
$\sin^2 25^\circ - \cos^2 65^\circ = \sin^2 25^\circ - \sin^2 (90^\circ - 65^\circ) = \sin^2 25^\circ - \sin^2 25^\circ = 0$.
If the question was indeed $\sin^2 25^\circ - \sin^2 65^\circ$, then the answer would be 0.
However, the question is explicitly written as $\sin^2 25^\circ + \sin^2 65^\circ$.
Let's assume there is a typo in the provided correct answer and proceed with the derived result.
The calculation is as follows:
We want to find the value of $\sin^2 25^\circ + \sin^2 65^\circ$.
We use the identity $\sin(90^\circ - \theta) = \cos \theta$.
Let $\theta = 25^\circ$. Then $90^\circ - \theta = 90^\circ - 25^\circ = 65^\circ$.
So, $\sin 65^\circ = \sin (90^\circ - 25^\circ) = \cos 25^\circ$.
Therefore, $\sin^2 65^\circ = (\cos 25^\circ)^2 = \cos^2 25^\circ$.
Substituting this into the expression:
$\sin^2 25^\circ + \sin^2 65^\circ = \sin^2 25^\circ + \cos^2 25^\circ$.
Using the Pythagorean identity $\sin^2 \phi + \cos^2 \phi = 1$ for any angle $\phi$, with $\phi = 25^\circ$, we get:
$\sin^2 25^\circ + \cos^2 25^\circ = 1$.
So, the value of $\sin^2 25^\circ + \sin^2 65^\circ$ is 1.
Given the provided correct answer is '0', and our calculation yields '1', there is a discrepancy. However, based on standard trigonometric identities, the sum is 1.
If we were forced to choose from the options and assume there was a reason for '0' to be correct, it would imply that the expression was meant to be $\sin^2 25^\circ - \sin^2 65^\circ$, or $\sin^2 25^\circ - \cos^2 65^\circ$, as shown above.
Let's assume the question is as stated, and the provided answer is incorrect. The correct answer should be 1.
However, I am asked to explain why the given correct answer (0) is correct. This means I must find a way to justify 0. This can only happen if there is a typo in the question.
Let's consider the possibility of a typo in the question that leads to the answer 0.
If the question was $\sin^2 25^\circ - \sin^2 65^\circ$, then:
$\sin^2 25^\circ - \sin^2 65^\circ = \sin^2 25^\circ - \cos^2 25^\circ$.
This is equal to $-(\cos^2 25^\circ - \sin^2 25^\circ) = -\cos(2 \times 25^\circ) = -\cos 50^\circ$. This is not 0.
If the question was $\sin^2 25^\circ - \cos^2 65^\circ$:
$\sin^2 25^\circ - \cos^2 65^\circ$. Since $\cos 65^\circ = \sin (90^\circ - 65^\circ) = \sin 25^\circ$, then $\cos^2 65^\circ = \sin^2 25^\circ$.
So, $\sin^2 25^\circ - \cos^2 65^\circ = \sin^2 25^\circ - \sin^2 25^\circ = 0$.
Given that the provided correct answer is 0, it is highly probable that the question was intended to be $\sin^2 25^\circ - \cos^2 65^\circ$ or $\sin^2 25^\circ - \sin^2 65^\circ$ where the second term is derived from $\sin 65^\circ = \cos 25^\circ$.
Assuming the question was meant to be $\sin^2 25^\circ - \cos^2 65^\circ$, here's the explanation:
We need to find the value of $\sin^2 25^\circ - \cos^2 65^\circ$.
We know that $\cos(90^\circ - \theta) = \sin \theta$.
Let $\theta = 65^\circ$. Then $90^\circ - \theta = 90^\circ - 65^\circ = 25^\circ$.
So, $\cos 65^\circ = \sin 25^\circ$.
Squaring both sides, we get $\cos^2 65^\circ = \sin^2 25^\circ$.
Substituting this into the expression:
$\sin^2 25^\circ - \cos^2 65^\circ = \sin^2 25^\circ - \sin^2 25^\circ = 0$.
However, since the problem explicitly states "Find the value of $\sin^2 25^\circ + \sin^2 65^\circ$", and the correct answer is given as '0', there is an inconsistency. Based on the stated problem, the answer should be 1. If the answer is indeed 0, then the problem statement must have a typo. Given the options, and that 0 is provided as correct, it is likely that the question was intended to be a subtraction.
If I must explain why 0 is the correct answer for the *given* question, it implies I have to force a logic that is not directly applicable. This is a problematic situation. I will proceed by highlighting the standard correct derivation for the given expression, and then point out the condition under which the answer would be 0, suggesting a likely typo in the problem statement if 0 is indeed the intended answer.
The problem asks for the value of $\sin^2 25^\circ + \sin^2 65^\circ$.
Using the complementary angle identity $\sin(90^\circ - \theta) = \cos \theta$, we have $\sin 65^\circ = \sin(90^\circ - 25^\circ) = \cos 25^\circ$.
Therefore, $\sin^2 65^\circ = \cos^2 25^\circ$.
Substituting this into the given expression:
$\sin^2 25^\circ + \sin^2 65^\circ = \sin^2 25^\circ + \cos^2 25^\circ$.
By the Pythagorean trigonometric identity, $\sin^2 \theta + \cos^2 \theta = 1$ for any angle $\theta$.
Thus, $\sin^2 25^\circ + \cos^2 25^\circ = 1$.
So, the value of $\sin^2 25^\circ + \sin^2 65^\circ$ is 1.
If the provided correct answer is 0, then the original question likely contained a typo and was intended to be $\sin^2 25^\circ - \cos^2 65^\circ$. In that case, since $\cos 65^\circ = \sin(90^\circ - 65^\circ) = \sin 25^\circ$, the expression becomes $\sin^2 25^\circ - \sin^2 25^\circ = 0$.
Given the instruction to explain why the answer '0' is correct, and the contradiction with the stated problem, I am unable to provide a valid mathematical explanation for the given problem to result in 0. The calculation for the stated problem clearly leads to 1.
The final answer is $\boxed{1}$.