If `x=cos^(-1)(cos4)" and "y=sin^(-1)(sin3)`, then which of the following conditions is true ?

Options

A. A) `tan(x-y)="tan"1/2`
B. B) `x+2y=2`
C. C) `tan(x+y)=-tan7`
D. D) `x+y+1=0`

Answer

To solve the problem, we need to evaluate the expressions for \( x \) and \( y \) given by:

1. \( x = \cos^{-1}(\cos 4) \)
2. \( y = \sin^{-1}(\sin 3) \)

### Step 1: Evaluate \( x \)

Given:
\[ x = \cos^{-1}(\cos 4) \]

The cosine function is periodic with a period of \( 2\pi \). The range of the \( \cos^{-1} \) function is \( [0, \pi] \). Since \( 4 \) radians is greater than \( \pi \), we can find an equivalent angle in the range \( [0, \pi] \).

Using the property:
\[ \cos(2\pi - \theta) = \cos(\theta) \]

We can rewrite \( \cos 4 \) as:
\[ \cos 4 = \cos(2\pi - 4) \]

Calculating \( 2\pi - 4 \):
\[ 2\pi - 4 \approx 2 \times 3.14 - 4 \approx 6.28 - 4 = 2.28 \]

Thus:
\[ x = 2\pi - 4 \]

### Step 2: Evaluate \( y \)

Given:
\[ y = \sin^{-1}(\sin 3) \]

The sine function is also periodic with a period of \( 2\pi \), and the range of the \( \sin^{-1} \) function is \( [-\frac{\pi}{2}, \frac{\pi}{2}] \). Since \( 3 \) radians is greater than \( \frac{\pi}{2} \), we can find an equivalent angle in the range \( [-\frac{\pi}{2}, \frac{\pi}{2}] \).

Using the property:
\[ \sin(\pi - \theta) = \sin(\theta) \]

We can rewrite \( \sin 3 \) as:
\[ \sin 3 = \sin(\pi - 3) \]

Calculating \( \pi - 3 \):
\[ \pi - 3 \approx 3.14 - 3 = 0.14 \]

Thus:
\[ y = \pi - 3 \]

### Step 3: Find the relationship between \( x \) and \( y \)

Now we have:
- \( x = 2\pi - 4 \)
- \( y = \pi - 3 \)

We can check the conditions given in the problem.

### Step 4: Check the conditions

Let's evaluate the expression \( \tan(x + y) \):

Calculating \( x + y \):
\[ x + y = (2\pi - 4) + (\pi - 3) = 3\pi - 7 \]

Now we need to evaluate \( \tan(3\pi - 7) \):
Using the property:
\[ \tan(3\pi - \theta) = -\tan(\theta) \]

Thus:
\[ \tan(3\pi - 7) = -\tan(7) \]

### Conclusion

After evaluating all options, we find that the correct condition that holds true is:

\[ \tan(x + y) = -\tan(7) \]