The value of `cos^(-1)(1/2)-2 sin^(-1)(1/2)+3cos^(-1)((-1)/(sqrt(2)))-4 tan^(-1)(-1)` is equal to

Options

A. `7 pi//4`
B. `13pi//4`
C. `pi//12`
D. `25pi//12`

Answer

To solve the expression \( \cos^{-1}\left(\frac{1}{2}\right) - 2 \sin^{-1}\left(\frac{1}{2}\right) + 3 \cos^{-1}\left(-\frac{1}{\sqrt{2}}\right) - 4 \tan^{-1}(-1) \), we will evaluate each term step by step.

### Step 1: Evaluate \( \cos^{-1}\left(\frac{1}{2}\right) \)
The value of \( \cos^{-1}\left(\frac{1}{2}\right) \) corresponds to the angle whose cosine is \( \frac{1}{2} \). This angle is:
\[ \cos^{-1}\left(\frac{1}{2}\right) = \frac{\pi}{3} \]

**Hint:** Recall that \( \cos\left(\frac{\pi}{3}\right) = \frac{1}{2} \).

### Step 2: Evaluate \( \sin^{-1}\left(\frac{1}{2}\right) \)
The value of \( \sin^{-1}\left(\frac{1}{2}\right) \) corresponds to the angle whose sine is \( \frac{1}{2} \). This angle is:
\[ \sin^{-1}\left(\frac{1}{2}\right) = \frac{\pi}{6} \]

**Hint:** Remember that \( \sin\left(\frac{\pi}{6}\right) = \frac{1}{2} \).

### Step 3: Evaluate \( 2 \sin^{-1}\left(\frac{1}{2}\right) \)
Now, we calculate:
\[ 2 \sin^{-1}\left(\frac{1}{2}\right) = 2 \times \frac{\pi}{6} = \frac{\pi}{3} \]

**Hint:** Multiplying the angle by 2 gives the new angle.

### Step 4: Evaluate \( \cos^{-1}\left(-\frac{1}{\sqrt{2}}\right) \)
The value of \( \cos^{-1}\left(-\frac{1}{\sqrt{2}}\right) \) corresponds to the angle whose cosine is \( -\frac{1}{\sqrt{2}} \). This angle is:
\[ \cos^{-1}\left(-\frac{1}{\sqrt{2}}\right) = \frac{3\pi}{4} \]

**Hint:** The cosine is negative in the second quadrant, where \( \cos\left(\frac{3\pi}{4}\right) = -\frac{1}{\sqrt{2}} \).

### Step 5: Evaluate \( 3 \cos^{-1}\left(-\frac{1}{\sqrt{2}}\right) \)
Now, we calculate:
\[ 3 \cos^{-1}\left(-\frac{1}{\sqrt{2}}\right) = 3 \times \frac{3\pi}{4} = \frac{9\pi}{4} \]

**Hint:** Again, multiply the angle by 3 to find the new angle.

### Step 6: Evaluate \( \tan^{-1}(-1) \)
The value of \( \tan^{-1}(-1) \) corresponds to the angle whose tangent is \( -1 \). This angle is:
\[ \tan^{-1}(-1) = -\frac{\pi}{4} \]

**Hint:** The tangent is negative in the fourth quadrant, where \( \tan\left(-\frac{\pi}{4}\right) = -1 \).

### Step 7: Evaluate \( -4 \tan^{-1}(-1) \)
Now, we calculate:
\[ -4 \tan^{-1}(-1) = -4 \times \left(-\frac{\pi}{4}\right) = \pi \]

**Hint:** Multiplying by -4 changes the sign and scales the angle.

### Step 8: Combine all the results
Now we combine all the evaluated parts:
\[ \cos^{-1}\left(\frac{1}{2}\right) - 2 \sin^{-1}\left(\frac{1}{2}\right) + 3 \cos^{-1}\left(-\frac{1}{\sqrt{2}}\right) - 4 \tan^{-1}(-1) \]
Substituting the values we found:
\[ \frac{\pi}{3} - \frac{\pi}{3} + \frac{9\pi}{4} + \pi \]

### Step 9: Simplify the expression
The first two terms cancel out:
\[ 0 + \frac{9\pi}{4} + \pi = \frac{9\pi}{4} + \frac{4\pi}{4} = \frac{13\pi}{4} \]

Thus, the final answer is:
\[ \boxed{\frac{13\pi}{4}} \]