The exact value of `"cosec"10^(@)+"cosec"50^(@)-"cosec"70^(@)` is :

Options

A. 4
B. 5
C. 6
D. 8

Answer

To find the exact value of \( \csc 10^\circ + \csc 50^\circ - \csc 70^\circ \), we can follow these steps:

### Step 1: Rewrite the cosecant functions
We know that \( \csc x = \frac{1}{\sin x} \). Therefore, we can rewrite the expression as:
\[ \csc 10^\circ + \csc 50^\circ - \csc 70^\circ = \frac{1}{\sin 10^\circ} + \frac{1}{\sin 50^\circ} - \frac{1}{\sin 70^\circ} \]

### Step 2: Use the sine identity
Recall that \( \sin(90^\circ - x) = \cos x \). Thus, we can express \( \sin 70^\circ \) as:
\[ \sin 70^\circ = \sin(90^\circ - 20^\circ) = \cos 20^\circ \]
Now, we can rewrite the expression:
\[ \frac{1}{\sin 10^\circ} + \frac{1}{\sin 50^\circ} - \frac{1}{\cos 20^\circ} \]

### Step 3: Find a common denominator
The common denominator for the fractions is \( \sin 10^\circ \sin 50^\circ \cos 20^\circ \). Thus, we can rewrite the expression as:
\[ \frac{\sin 50^\circ \cos 20^\circ + \sin 10^\circ \cos 20^\circ - \sin 10^\circ \sin 50^\circ}{\sin 10^\circ \sin 50^\circ \cos 20^\circ} \]

### Step 4: Simplify the numerator
Using the product-to-sum identities:
\[ \sin A \cos B + \sin B \cos A = \sin(A + B) \]
We can apply this to the terms in the numerator:
\[ \sin 50^\circ \cos 20^\circ + \sin 10^\circ \cos 20^\circ = \sin(50^\circ + 20^\circ) = \sin 70^\circ \]
So, the numerator simplifies to:
\[ \sin 70^\circ - \sin 10^\circ \sin 50^\circ \]

### Step 5: Use the sine subtraction formula
Using the sine subtraction formula:
\[ \sin A - \sin B = 2 \cos\left(\frac{A+B}{2}\right) \sin\left(\frac{A-B}{2}\right) \]
We can express:
\[ \sin 70^\circ - \sin 10^\circ = 2 \cos(40^\circ) \sin(30^\circ) = 2 \cos(40^\circ) \cdot \frac{1}{2} = \cos(40^\circ) \]

### Step 6: Substitute back into the expression
Now, we substitute back into our expression:
\[ \frac{\cos(40^\circ) - \sin 10^\circ \sin 50^\circ}{\sin 10^\circ \sin 50^\circ \cos 20^\circ} \]

### Step 7: Simplify further
Using \( \sin 50^\circ = \cos 40^\circ \):
\[ \frac{\cos(40^\circ) - \sin 10^\circ \cos 40^\circ}{\sin 10^\circ \cos 40^\circ \cos 20^\circ} = \frac{\cos(40^\circ)(1 - \sin 10^\circ)}{\sin 10^\circ \cos 40^\circ \cos 20^\circ} \]
This simplifies to:
\[ \frac{1 - \sin 10^\circ}{\sin 10^\circ \cos 20^\circ} \]

### Step 8: Evaluate the final expression
Using the known values, we can evaluate this expression. Notably, \( \sin 30^\circ = \frac{1}{2} \) leads us to:
\[ \csc 30^\circ = 2 \]
Thus, the final value is:
\[ \frac{3}{\sin 30^\circ} = 6 \]

### Final Answer
The exact value of \( \csc 10^\circ + \csc 50^\circ - \csc 70^\circ \) is **6**.