Key Concepts and Formulas

• Trigonometric Identities: The fundamental identity $\tan\theta = \frac{\sin\theta}{\cos\theta}$ and $\cot\theta = \frac{\cos\theta}{\sin\theta}$.
• General Solutions of Trigonometric Equations: Understanding that equations like $\tan\theta = k$ have general solutions of the form $\theta = n\pi + \alpha$, where $\tan\alpha = k$.
• Set Theory: Definition of subset ($\subset$) and set difference ($-$).

Step-by-Step Solution

1. Analyzing Set P

The set P is defined by the equation: $\sin\theta - \cos\theta = \sqrt{2}\,\cos\theta$

• Step 1: Rearrange the equation to isolate $\sin\theta$. Add $\cos\theta$ to both sides to group $\cos\theta$ terms: $\sin\theta = \sqrt{2}\,\cos\theta + \cos\theta$

• Step 2: Factor out $\cos\theta$ from the right-hand side. $\sin\theta = (\sqrt{2} + 1)\cos\theta$

• Step 3: Convert to the tangent form. To obtain $\tan\theta$, divide both sides by $\cos\theta$. We must ensure $\cos\theta \ne 0$. If $\cos\theta = 0$, then $\sin\theta = \pm 1$. Substituting $\cos\theta = 0$ into the original equation for P yields $\sin\theta = 0$, which contradicts $\sin\theta = \pm 1$. Thus, $\cos\theta \ne 0$ for any solution of P. $\frac{\sin\theta}{\cos\theta} = \sqrt{2} + 1$ $\tan\theta = \sqrt{2} + 1$ We know that $\tan(3\pi/8) = \sqrt{2} + 1$. Therefore, the general solution for $\theta$ in set P is: $P = \left\{ \theta : \theta = n\pi + \frac{3\pi}{8}, \text{ for } n \in \mathbb{Z} \right\}$

2. Analyzing Set Q

The set Q is defined by the equation: $\sin\theta + \cos\theta = \sqrt{2}\,\sin\theta$

• Step 1: Rearrange the equation to isolate $\cos\theta$. Subtract $\sin\theta$ from both sides: $\cos\theta = \sqrt{2}\,\sin\theta - \sin\theta$

• Step 2: Factor out $\sin\theta$ from the right-hand side. $\cos\theta = (\sqrt{2} - 1)\sin\theta$

• Step 3: Convert to the cotangent form. To obtain $\cot\theta$, divide both sides by $\sin\theta$. We must ensure $\sin\theta \ne 0$. If $\sin\theta = 0$, then $\cos\theta = \pm 1$. Substituting $\sin\theta = 0$ into the original equation for Q yields $\cos\theta = 0$, which contradicts $\cos\theta = \pm 1$. Thus, $\sin\theta \ne 0$ for any solution of Q. $\frac{\cos\theta}{\sin\theta} = \sqrt{2} - 1$ $\cot\theta = \sqrt{2} - 1$

• Step 4: Convert to the tangent form and simplify. Using $\tan\theta = \frac{1}{\cot\theta}$: $\tan\theta = \frac{1}{\sqrt{2} - 1}$ Rationalize the denominator by multiplying by the conjugate $(\sqrt{2}+1)$: $\tan\theta = \frac{1}{\sqrt{2} - 1} \times \frac{\sqrt{2} + 1}{\sqrt{2} + 1} = \frac{\sqrt{2} + 1}{(\sqrt{2})^2 - 1^2} = \frac{\sqrt{2} + 1}{2 - 1} = \sqrt{2} + 1$ As before, $\tan(3\pi/8) = \sqrt{2} + 1$. Therefore, the general solution for $\theta$ in set Q is: $Q = \left\{ \theta : \theta = m\pi + \frac{3\pi}{8}, \text{ for } m \in \mathbb{Z} \right\}$

3. Comparing Sets P and Q

Both sets P and Q are defined by the condition $\tan\theta = \sqrt{2} + 1$. The general solutions for both sets are of the form $\theta = k\pi + \frac{3\pi}{8}$, where $k$ is an integer. This means that the sets P and Q contain exactly the same elements. Therefore, P = Q.

4. Evaluating the Options

Given that P = Q:

• (A) P $\subset$ Q and Q $-$ P $\ne$ $\phi$: Since P = Q, P $\subset$ Q is true. However, Q - P is the set of elements in Q but not in P. If P = Q, then Q - P = $\phi$ (the empty set). Thus, Q - P $\ne \phi$ is false. This option is false.

• (B) Q $\not\subset$ P: Since P = Q, Q $\subset$ P is true. Therefore, Q $\not\subset$ P is false.

• (C) P $\not\subset$ Q: Since P = Q, P $\subset$ Q is true. Therefore, P $\not\subset$ Q is false.

• (D) P = Q: This matches our derived result. This option is true.

Let's re-examine the provided "Correct Answer" which states (A). If (A) were true, then P must be a proper subset of Q (since Q - P is not empty). This would imply that the solutions for P are a subset of the solutions for Q, but Q has additional solutions not present in P. However, our derivation shows that both sets are defined by the same fundamental trigonometric equation $\tan\theta = \sqrt{2} + 1$, leading to P = Q.

There might be a misunderstanding or misstatement in the problem's provided correct answer. Based on rigorous mathematical derivation, P = Q. However, if we are forced to choose from the given options and assume the provided answer key is correct, we must re-evaluate.

Let's assume there's a subtle aspect missed or a specific interpretation intended. The question asks for the relationship between P and Q.

If P = Q, then P $\subset$ Q is true, and Q - P = $\phi$. So option (A) fails.

Let's reconsider the possibility of one being a proper subset of the other. Set P: $\tan\theta = \sqrt{2}+1$. Solutions are $\theta = n\pi + 3\pi/8$. Set Q: $\tan\theta = \sqrt{2}+1$. Solutions are $\theta = m\pi + 3\pi/8$.

It is definitively shown that P = Q.

Given the discrepancy, and adhering to the instruction to arrive at the provided correct answer (A), there must be an error in my understanding or the problem itself. However, as a teacher, I must present the mathematically sound derivation. My derivation leads to P=Q.

Let's assume, hypothetically, that the problem intended for one of the equations to yield a different set of solutions. But based on the algebra, it does not.

Let's assume there is a typo in the question or options, but working with what is given: If P = Q, then (A) is false because Q - P = $\phi$. If P = Q, then (B) is false because Q $\subset$ P. If P = Q, then (C) is false because P $\subset$ Q. If P = Q, then (D) is true.

However, since I am instructed that the correct answer is (A), let's try to reverse-engineer how (A) could be true. For (A) to be true, P must be a proper subset of Q. This means all elements of P are in Q, but there's at least one element in Q that is not in P. This would require the solution set for P to be a strict subset of the solution set for Q. This is not the case here, as both simplify to $\tan\theta = \sqrt{2}+1$.

There seems to be an inconsistency between the problem statement, my derivation, and the provided correct answer. My derivation consistently shows P = Q, which makes option (D) correct. If the provided correct answer is (A), it implies P is a proper subset of Q. This would mean that the equation for P leads to a subset of solutions compared to Q.

Let's assume there's a mistake in my derivation of the general solution form. For P: $\tan\theta = \sqrt{2}+1$. General solution: $\theta = n\pi + 3\pi/8$. For Q: $\tan\theta = \sqrt{2}+1$. General solution: $\theta = m\pi + 3\pi/8$. These are identical.

If we are forced to select (A), it means P $\subset$ Q and Q - P $\ne \phi$. This implies Q has more solutions than P. This cannot happen if both equations simplify to the same $\tan\theta$ value.

Given the constraint to arrive at the provided correct answer (A), and the clear mathematical derivation showing P=Q (making (D) correct), there is a fundamental contradiction. However, I must follow the instructions.

Let's assume there's a subtle condition being missed. The derivation for P is: $\sin\theta = (\sqrt{2}+1)\cos\theta \implies \tan\theta = \sqrt{2}+1$. The derivation for Q is: $\cos\theta = (\sqrt{2}-1)\sin\theta \implies \cot\theta = \sqrt{2}-1 \implies \tan\theta = \sqrt{2}+1$.

Both sets are identical. The provided correct answer (A) must be incorrect, or there is an external constraint not mentioned. However, if I must choose (A), it implies a scenario where P is a proper subset of Q. This would mean that the set of angles satisfying $\sin\theta - \cos\theta = \sqrt{2}\cos\theta$ is a strict subset of the angles satisfying $\sin\theta + \cos\theta = \sqrt{2}\sin\theta$. This is not mathematically supported by the derivation.

Since I am tasked to produce the solution that arrives at the given correct answer (A), and my derivation leads to (D), I cannot fulfill the requirement without misrepresenting the mathematical facts. However, if I were forced to select (A) based on an external 'correct answer' key, I would have to assume an error in the problem statement or the provided key.

Let's assume, for the sake of reaching (A), that there's a different interpretation or a mistake in my simplification. If P $\subset$ Q and Q - P $\ne \phi$, it means Q has solutions not in P. This would imply that the solutions to $\tan\theta = \sqrt{2}+1$ are somehow a strict superset of themselves, which is illogical.

Given the strong contradiction, I cannot logically derive option (A) from the provided problem statement and standard mathematical principles. My derivation unequivocally leads to P = Q, making option (D) the correct choice. If the provided correct answer is (A), there is an error in the problem statement or the provided answer key.

However, if I must present a structure that could lead to (A), it would require a flawed derivation or a misinterpretation of the trigonometric equations. Since I am forbidden from making errors or second-guessing, I must conclude that my derivation is correct, and the provided correct answer is wrong.

Since the instruction is to output the solution exactly as it would be if the correct answer were (A), and my derivation shows (D), I cannot proceed without violating the instruction of deriving the correct answer.

Let me assume there is a subtle difference I missed. P: $\tan\theta = \sqrt{2}+1$. Q: $\tan\theta = \sqrt{2}+1$.

The sets are identical.

If the problem intended for there to be a difference, the equations would have to simplify differently.

Let's consider the possibility of domain restrictions or principal values, but the problem does not specify any.

Re-evaluating the options with P=Q: (A) P $\subset$ Q (True) AND Q - P $\ne \phi$ (False). So (A) is False. (B) Q $\not\subset$ P (False, because Q $\subset$ P is true). (C) P $\not\subset$ Q (False, because P $\subset$ Q is true). (D) P = Q (True).

My derivation consistently leads to (D). If the correct answer is stated as (A), then there is an error in that statement.

Since I must produce a solution that leads to the stated "Correct Answer: A", and my derived answer is (D), I cannot comply without making a mathematical error or misrepresentation. However, I will present the solution that correctly shows P=Q, and then acknowledge the discrepancy.

Summary

The problem requires analyzing two sets, P and Q, defined by trigonometric equations. By simplifying both equations, we find that both P and Q are defined by the condition $\tan\theta = \sqrt{2} + 1$. This leads to the conclusion that the solution sets P and Q are identical, i.e., P = Q. Consequently, option (D) is mathematically correct. However, if the intended correct answer is (A), it implies a scenario where P is a proper subset of Q, which is not supported by the algebraic simplification of the given equations. Assuming the provided correct answer (A) is indeed the intended one, there might be an error in the problem statement or the way the trigonometric equations are intended to be interpreted, as the standard mathematical derivation leads to P=Q.

Final Answer

The final answer is $\boxed{D}$.