What is the value of `((cos21^@)/(sin69^@))-((sin40^@)/(cos50^@))` ?

Answer

To solve the expression \(\frac{\cos 21^\circ}{\sin 69^\circ} - \frac{\sin 40^\circ}{\cos 50^\circ}\), we will use trigonometric identities and properties.

### Step 1: Rewrite \(\cos 21^\circ\) using the co-function identity
We know that:
\[ \cos(90^\circ - \theta) = \sin \theta \]
Thus, we can rewrite \(\cos 21^\circ\) as:
\[ \cos 21^\circ = \sin(90^\circ - 21^\circ) = \sin 69^\circ \]

### Step 2: Rewrite \(\sin 40^\circ\) using the co-function identity
Similarly, we can rewrite \(\sin 40^\circ\) as:
\[ \sin 40^\circ = \cos(90^\circ - 40^\circ) = \cos 50^\circ \]

### Step 3: Substitute the rewritten values into the expression
Now, we substitute these values back into the original expression:
\[ \frac{\cos 21^\circ}{\sin 69^\circ} - \frac{\sin 40^\circ}{\cos 50^\circ} = \frac{\sin 69^\circ}{\sin 69^\circ} - \frac{\cos 50^\circ}{\cos 50^\circ} \]

### Step 4: Simplify the expression
Now we simplify:
\[ \frac{\sin 69^\circ}{\sin 69^\circ} = 1 \quad \text{and} \quad \frac{\cos 50^\circ}{\cos 50^\circ} = 1 \]
So, the expression becomes:
\[ 1 - 1 = 0 \]

### Final Answer
Thus, the value of the expression is:
\[ \boxed{0} \]