If `5sin x =4`, then the numberical value of `((tanx-cotx)/(secx-tanx))((cos^(4)x-sin^(4)x)/(2cos^(2)x-1))` ?

Options

A. `(3)/(5)`
B. `(5)/(4)`
C. `(7)/(4)`
D. `(9)/(5)`

Answer

To solve the problem, we need to find the value of the expression

\[ \frac{(\tan x - \cot x)}{(\sec x - \tan x)} \cdot \frac{(\cos^4 x - \sin^4 x)}{(2 \cos^2 x - 1)} \]

given that \(5 \sin x = 4\).

### Step 1: Find \(\sin x\) and \(\cos x\)

From the equation \(5 \sin x = 4\), we can find \(\sin x\):

\[ \sin x = \frac{4}{5} \]

Next, we can find \(\cos x\) using the Pythagorean identity:

\[ \cos^2 x = 1 - \sin^2 x = 1 - \left(\frac{4}{5}\right)^2 = 1 - \frac{16}{25} = \frac{9}{25} \]

Thus,

\[ \cos x = \sqrt{\frac{9}{25}} = \frac{3}{5} \]

### Step 2: Calculate \(\tan x\) and \(\cot x\)

Now we can find \(\tan x\) and \(\cot x\):

\[ \tan x = \frac{\sin x}{\cos x} = \frac{\frac{4}{5}}{\frac{3}{5}} = \frac{4}{3} \]

\[ \cot x = \frac{1}{\tan x} = \frac{3}{4} \]

### Step 3: Calculate \(\sec x\)

Next, we find \(\sec x\):

\[ \sec x = \frac{1}{\cos x} = \frac{5}{3} \]

### Step 4: Substitute values into the expression

Now we substitute these values into the expression:

1. Calculate \(\tan x - \cot x\):

\[ \tan x - \cot x = \frac{4}{3} - \frac{3}{4} = \frac{16}{12} - \frac{9}{12} = \frac{7}{12} \]

2. Calculate \(\sec x - \tan x\):

\[ \sec x - \tan x = \frac{5}{3} - \frac{4}{3} = \frac{1}{3} \]

3. Calculate \(\cos^4 x - \sin^4 x\):

Using the difference of squares:

\[ \cos^4 x - \sin^4 x = (\cos^2 x - \sin^2 x)(\cos^2 x + \sin^2 x) \]

Since \(\cos^2 x + \sin^2 x = 1\):

\[ \cos^4 x - \sin^4 x = \cos^2 x - \sin^2 x = \frac{9}{25} - \frac{16}{25} = -\frac{7}{25} \]

4. Calculate \(2 \cos^2 x - 1\):

\[ 2 \cos^2 x - 1 = 2 \cdot \frac{9}{25} - 1 = \frac{18}{25} - \frac{25}{25} = -\frac{7}{25} \]

### Step 5: Substitute into the expression

Now we substitute everything back into the expression:

\[ \frac{\left(\frac{7}{12}\right)}{\left(\frac{1}{3}\right)} \cdot \frac{\left(-\frac{7}{25}\right)}{\left(-\frac{7}{25}\right)} \]

This simplifies to:

\[ \frac{7}{12} \cdot 3 = \frac{21}{12} = \frac{7}{4} \]

### Final Answer

The value of the expression is

\[ \frac{7}{4} \]