The slope of the tangent at `(x , y)` to a curve passing through `(1,pi/4)` is given by `y/x-cos^2(y/x),` then the equation of the curve is (a) `( b ) (c) y=( d ) (e)tan^(( f ) (g)-1( h ))( i )(( j ) (k)log(( l ) (m) (n) e/( o ) x (p) (q) (r))( s ))( t )` (u) (v) `( w ) (x) y=x (y) (z)tan^(( a a ) (bb)-1( c c ))( d d )(( e e ) (ff)log(( g g ) (hh) (ii) x/( j j ) e (kk) (ll) (mm))( n n ))( o o )` (pp) (qq) `( r r ) (ss) y=x (tt) (uu)tan^(( v v ) (ww)-1( x x ))( y y )(( z z ) (aaa)log(( b b b ) (ccc) (ddd) e/( e e e ) x (fff) (ggg) (hhh))( i i i ))( j j j )` (kkk) (d) none of these

Options

A. `y=tan^(-1){log((e)/(x))}`
B. `y=x tan^(-1){log((x)/(e))}`
C. `y=x tan^(-1){log((e)/(x))}`
D. none of these

Answer

To solve the problem, we need to find the equation of the curve given the slope of the tangent at any point \((x, y)\) on the curve as:

\[ \frac{dy}{dx} = \frac{y}{x} - \cos^2\left(\frac{y}{x}\right) \]

The curve passes through the point \((1, \frac{\pi}{4})\).

### Step 1: Rewrite the equation
We start by rewriting the differential equation:

\[ \frac{dy}{dx} = \frac{y}{x} - \cos^2\left(\frac{y}{x}\right) \]

### Step 2: Homogeneous substitution
Since this is a homogeneous equation, we can use the substitution \(y = vx\), where \(v = \frac{y}{x}\). Then, we differentiate \(y\) with respect to \(x\):

\[ \frac{dy}{dx} = v + x\frac{dv}{dx} \]

### Step 3: Substitute into the differential equation
Substituting \(y = vx\) into the original equation gives:

\[ v + x\frac{dv}{dx} = v - \cos^2(v) \]

### Step 4: Simplify the equation
Now, we can cancel \(v\) from both sides:

\[ x\frac{dv}{dx} = -\cos^2(v) \]

### Step 5: Separate variables
We can separate the variables:

\[ \frac{dv}{\cos^2(v)} = -\frac{dx}{x} \]

### Step 6: Integrate both sides
Integrating both sides, we have:

\[ \int \sec^2(v) dv = -\int \frac{dx}{x} \]

The left side integrates to:

\[ \tan(v) = -\log(x) + C \]

### Step 7: Substitute back for \(v\)
Recall that \(v = \frac{y}{x}\), so we substitute back:

\[ \tan\left(\frac{y}{x}\right) = -\log(x) + C \]

### Step 8: Use the initial condition
We use the initial condition that the curve passes through \((1, \frac{\pi}{4})\):

\[ \tan\left(\frac{\pi/4}{1}\right) = -\log(1) + C \]

Since \(\tan\left(\frac{\pi}{4}\right) = 1\) and \(\log(1) = 0\), we have:

\[ 1 = 0 + C \implies C = 1 \]

### Step 9: Final equation of the curve
Substituting \(C\) back into the equation gives:

\[ \tan\left(\frac{y}{x}\right) = -\log(x) + 1 \]

### Step 10: Rearranging the equation
Rearranging gives:

\[ \tan\left(\frac{y}{x}\right) = 1 - \log(x) \]

### Step 11: Solve for \(y\)
Finally, we can express \(y\) in terms of \(x\):

\[ \frac{y}{x} = \tan^{-1}(1 - \log(x)) \implies y = x \tan^{-1}(1 - \log(x)) \]

### Conclusion
Thus, the equation of the curve is:

\[ y = x \tan^{-1}(1 - \log(x)) \]