If the dependent variable y is changed to z by the substitution method y=tanz then the differential equation `d^2y/dx^2=1+2(1+y)/(1+y^2)(dy/dx)^2` is changed to `d^2z/dx^2=cos^2z+k(dz/dx)^2` then find the value off k

Explanation

To solve the problem, we will follow these steps:

### Step 1: Substitute the variable
We start with the substitution \( y = \tan z \).

### Step 2: Find the first derivative
We differentiate \( y \) with respect to \( x \):
\[ \frac{dy}{dx} = \sec^2 z \cdot \frac{dz}{dx} \]

### Step 3: Find the second derivative
Next, we differentiate \( \frac{dy}{dx} \) to find \( \frac{d^2y}{dx^2} \):
\[ \frac{d^2y}{dx^2} = \frac{d}{dx} \left( \sec^2 z \cdot \frac{dz}{dx} \right) \]
Using the product rule, we have:
\[ \frac{d^2y}{dx^2} = \frac{d}{dx}(\sec^2 z) \cdot \frac{dz}{dx} + \sec^2 z \cdot \frac{d^2z}{dx^2} \]
The derivative of \( \sec^2 z \) is \( 2 \sec^2 z \tan z \cdot \frac{dz}{dx} \), so:
\[ \frac{d^2y}{dx^2} = 2 \sec^2 z \tan z \left( \frac{dz}{dx} \right)^2 + \sec^2 z \frac{d^2z}{dx^2} \]

### Step 4: Substitute into the original differential equation
The original equation is:
\[ \frac{d^2y}{dx^2} = 1 + \frac{2(1+y)}{(1+y^2)} \left( \frac{dy}{dx} \right)^2 \]
Substituting \( y = \tan z \) gives:
\[ \frac{d^2y}{dx^2} = 1 + \frac{2(1+\tan z)}{(1+\tan^2 z)} \left( \sec^2 z \cdot \frac{dz}{dx} \right)^2 \]
Since \( 1 + \tan^2 z = \sec^2 z \), we can simplify:
\[ \frac{d^2y}{dx^2} = 1 + 2(1+\tan z) \cdot \frac{(\sec^2 z \cdot \frac{dz}{dx})^2}{\sec^2 z} \]
This simplifies to:
\[ \frac{d^2y}{dx^2} = 1 + 2(1+\tan z) \sec^2 z \left( \frac{dz}{dx} \right)^2 \]

### Step 5: Set the equations equal
Now we equate the two expressions for \( \frac{d^2y}{dx^2} \):
\[ 2 \sec^2 z \tan z \left( \frac{dz}{dx} \right)^2 + \sec^2 z \frac{d^2z}{dx^2} = 1 + 2(1+\tan z) \sec^2 z \left( \frac{dz}{dx} \right)^2 \]

### Step 6: Rearranging the equation
Rearranging gives:
\[ \sec^2 z \frac{d^2z}{dx^2} = 1 + 2(1+\tan z) \sec^2 z \left( \frac{dz}{dx} \right)^2 - 2 \sec^2 z \tan z \left( \frac{dz}{dx} \right)^2 \]
This simplifies to:
\[ \sec^2 z \frac{d^2z}{dx^2} = 1 + 2 \sec^2 z \left( \frac{dz}{dx} \right)^2 \]

### Step 7: Final form
Dividing through by \( \sec^2 z \):
\[ \frac{d^2z}{dx^2} = \cos^2 z + k \left( \frac{dz}{dx} \right)^2 \]
Comparing this with the given form \( \frac{d^2z}{dx^2} = \cos^2 z + k \left( \frac{dz}{dx} \right)^2 \), we find that:
\[ k = 2 \]

### Final Answer
Thus, the value of \( k \) is:
\[ \boxed{2} \]