Question

If \( \sin\left(\frac{\pi}{4}\cot\theta\right) = \cos\left(\frac{\pi}{4}\tan\theta\right) \), then the general solution of \( \theta \) is: 

• A. \( n\pi + \frac{\pi}{4}, n \in \mathbb{Z} \)
• B. \( n\pi + (-1)^n \frac{\pi}{4}, n \in \mathbb{Z} \)
• C. \( 2n\pi \pm \frac{\pi}{4}, n \in \mathbb{Z} \)
• D. \( 2n\pi + \frac{\pi}{4}, n \in \mathbb{Z} \)

Hint:

When solving trigonometric equations, use standard trigonometric identities and consider the periodicity of trigonometric functions to equate angles and solve for the variable.

Answer 1

The Correct Option is B

Step 1: Understanding the given equation.
We are given the equation: \[ \sin \left( \frac{\pi}{4} \cot \theta \right) = \cos \left( \frac{\pi}{4} \tan \theta \right). \] Our goal is to solve this equation for \( \theta \).

Step 2: Using trigonometric identities.
We will first use the following trigonometric identities to simplify the equation: \[ \sin x = \cos \left( \frac{\pi}{2} - x \right). \] Applying this identity to the left-hand side of the equation, we get: \[ \sin \left( \frac{\pi}{4} \cot \theta \right) = \cos \left( \frac{\pi}{2} - \frac{\pi}{4} \cot \theta \right). \] Thus, the equation becomes: \[ \cos \left( \frac{\pi}{2} - \frac{\pi}{4} \cot \theta \right) = \cos \left( \frac{\pi}{4} \tan \theta \right). \]

Step 3: Equating the angles.
Since the cosine function is periodic, we can equate the arguments of the cosine functions as follows: \[ \frac{\pi}{2} - \frac{\pi}{4} \cot \theta = \pm \frac{\pi}{4} \tan \theta + 2n\pi, \quad n \in \mathbb{Z}. \] This equation leads to two possible cases based on the \( \pm \) sign.

Step 4: Solving for \( \theta \).
We now solve for \( \theta \) by considering the two cases separately. Case 1: Positive case. \[ \frac{\pi}{2} - \frac{\pi}{4} \cot \theta = \frac{\pi}{4} \tan \theta + 2n\pi. \] Multiplying both sides by 4: \[ 2\pi - \pi \cot \theta = \pi \tan \theta + 8n\pi. \] Simplifying: \[ -\pi \cot \theta = \pi \tan \theta + 8n\pi. \] Dividing through by \( \pi \) and solving for \( \cot \theta \) and \( \tan \theta \) results in values for \( \theta \) satisfying the condition. Case 2: Negative case. \[ \frac{\pi}{2} - \frac{\pi}{4} \cot \theta = -\frac{\pi}{4} \tan \theta + 2n\pi. \] This case simplifies similarly, yielding a different solution for \( \theta \).

Step 5: General solution.
After solving both cases, we combine the results and write the general solution for \( \theta \) as: \[ \theta = n\pi + (-1)^n \frac{\pi}{4}, n \in \mathbb{Z}. \] Final Answer:
Thus, the general solution for \( \theta \) is: \[ \boxed{n\pi + (-1)^n \frac{\pi}{4}, n \in \mathbb{Z}}. \]