If `6(sec^(2) 59^(@)-cot^(2)31^(@))-(2)/(3) sin90^(@)-3 tan^(2)56^(@)y tan^(2)34^(@)=(y)/(3)` then the value of y is :

Options

A. `(8)/(5)`
B. `-(8)/(5)`
C. `(2)/(3)`
D. `(-2)/(3)`

Answer

To solve the equation given in the problem, we will follow a step-by-step approach:

### Step 1: Rewrite the equation
The equation is given as:
\[ 6(\sec^2 59^\circ - \cot^2 31^\circ) - \frac{2}{3} \sin 90^\circ - 3 \tan^2 56^\circ y \tan^2 34^\circ = \frac{y}{3} \]

### Step 2: Simplify \(\sin 90^\circ\)
We know that \(\sin 90^\circ = 1\). Thus, we can rewrite the equation:
\[ 6(\sec^2 59^\circ - \cot^2 31^\circ) - \frac{2}{3} - 3 \tan^2 56^\circ y \tan^2 34^\circ = \frac{y}{3} \]

### Step 3: Use trigonometric identities
Using the identity \(\sec^2 \theta - \cot^2 \theta = 1\):
\[ \sec^2 59^\circ - \cot^2 31^\circ = 1 \]
So, we can substitute this into the equation:
\[ 6(1) - \frac{2}{3} - 3 \tan^2 56^\circ y \tan^2 34^\circ = \frac{y}{3} \]

### Step 4: Simplify further
This simplifies to:
\[ 6 - \frac{2}{3} - 3 \tan^2 56^\circ y \tan^2 34^\circ = \frac{y}{3} \]

### Step 5: Find a common denominator
To combine the constants, we convert \(6\) into a fraction with a denominator of \(3\):
\[ 6 = \frac{18}{3} \]
Thus, we have:
\[ \frac{18}{3} - \frac{2}{3} - 3 \tan^2 56^\circ y \tan^2 34^\circ = \frac{y}{3} \]

### Step 6: Combine the fractions
Combining the fractions gives:
\[ \frac{16}{3} - 3 \tan^2 56^\circ y \tan^2 34^\circ = \frac{y}{3} \]

### Step 7: Clear the fractions
Multiply the entire equation by \(3\) to eliminate the denominators:
\[ 16 - 9 \tan^2 56^\circ y \tan^2 34^\circ = y \]

### Step 8: Rearrange the equation
Rearranging gives:
\[ 16 = y + 9 \tan^2 56^\circ y \tan^2 34^\circ \]
Factoring out \(y\):
\[ 16 = y(1 + 9 \tan^2 56^\circ \tan^2 34^\circ) \]

### Step 9: Solve for \(y\)
Now, we can solve for \(y\):
\[ y = \frac{16}{1 + 9 \tan^2 56^\circ \tan^2 34^\circ} \]

### Step 10: Calculate the value of \(y\)
To find the exact value of \(y\), we need to calculate \(9 \tan^2 56^\circ \tan^2 34^\circ\). Using the values:
- \(\tan 56^\circ \approx 1.4826\)
- \(\tan 34^\circ \approx 0.6745\)

Calculating:
\[ \tan^2 56^\circ \approx 2.198 \]
\[ \tan^2 34^\circ \approx 0.454 \]
\[ 9 \tan^2 56^\circ \tan^2 34^\circ \approx 9 \times 2.198 \times 0.454 \approx 8.999 \]

Thus,
\[ y \approx \frac{16}{1 + 8.999} = \frac{16}{9.999} \approx 1.6 \]

### Final Value of \(y\)
After simplification, we find:
\[ y = \frac{16}{10} = \frac{8}{5} \]

### Conclusion
The value of \(y\) is \(\frac{8}{5}\).