If cos x= tan `40^@ tan 50^@` , what is the value of x ?

Answer

To solve the equation \( \cos x = \tan 40^\circ \tan 50^\circ \), we can follow these steps:

### Step 1: Use the identity for tangent
We know that:
\[ \tan A \tan B = \frac{\sin A \sin B}{\cos A \cos B} \]
For \( A = 40^\circ \) and \( B = 50^\circ \), we can use the identity that states:
\[ \tan 40^\circ \tan 50^\circ = 1 \]
This is because \( 40^\circ + 50^\circ = 90^\circ \), and the tangent of complementary angles is equal to 1.

### Step 2: Substitute back into the equation
Now substituting this back into our equation:
\[ \cos x = 1 \]

### Step 3: Find the angle x
The cosine of an angle is equal to 1 at:
\[ x = 0^\circ + 360^\circ n \quad (n \in \mathbb{Z}) \]
For the principal value, we take \( n = 0 \):
\[ x = 0^\circ \]

### Final Answer
Thus, the value of \( x \) is:
\[ \boxed{0^\circ} \]