The value of `(cos^(3)20^(@)-cos^(3)70^(@))/(sin^(3)70^(@)-sin^(3)20^(@))=?`

Options

A. 5
B. 7
C. 1
D. 3

Answer

To solve the expression \((\cos^3 20^\circ - \cos^3 70^\circ) / (\sin^3 70^\circ - \sin^3 20^\circ)\), we can follow these steps:

### Step 1: Rewrite the cosine terms using the sine identity
We know that \(\cos \theta = \sin(90^\circ - \theta)\). Therefore, we can rewrite \(\cos 20^\circ\) and \(\cos 70^\circ\):
\[ \cos 20^\circ = \sin(90^\circ - 20^\circ) = \sin 70^\circ \]
\[ \cos 70^\circ = \sin(90^\circ - 70^\circ) = \sin 20^\circ \]

### Step 2: Substitute these identities into the expression
Now we can substitute these identities into the original expression:
\[ \frac{\cos^3 20^\circ - \cos^3 70^\circ}{\sin^3 70^\circ - \sin^3 20^\circ} = \frac{\sin^3 70^\circ - \sin^3 20^\circ}{\sin^3 70^\circ - \sin^3 20^\circ} \]

### Step 3: Simplify the expression
Since the numerator and denominator are now the same, we can simplify:
\[ \frac{\sin^3 70^\circ - \sin^3 20^\circ}{\sin^3 70^\circ - \sin^3 20^\circ} = 1 \]

### Final Answer
Thus, the value of the expression is:
\[ \boxed{1} \]