The value of `tan80^(@)tan10^(@)+sin^(2)70^(@)+sin^(2)20^(@)` is

Options

A. 0
B. 1
C. 2
D. `(sqrt(3))/(2)`

Answer

To solve the expression \( \tan 80^\circ \tan 10^\circ + \sin^2 70^\circ + \sin^2 20^\circ \), we will break it down into manageable parts.

### Step 1: Simplify \( \tan 80^\circ \tan 10^\circ \)

Using the identity \( \tan(90^\circ - x) = \cot x \), we can rewrite \( \tan 80^\circ \) as:
\[ \tan 80^\circ = \cot 10^\circ \]
Thus, we can express the product:
\[ \tan 80^\circ \tan 10^\circ = \cot 10^\circ \tan 10^\circ = 1 \]

### Step 2: Simplify \( \sin^2 70^\circ \)

Using the identity \( \sin(90^\circ - x) = \cos x \), we can rewrite \( \sin 70^\circ \) as:
\[ \sin 70^\circ = \cos 20^\circ \]
Therefore, we have:
\[ \sin^2 70^\circ = \cos^2 20^\circ \]

### Step 3: Simplify \( \sin^2 20^\circ \)

This term remains as is:
\[ \sin^2 20^\circ \]

### Step 4: Combine the results

Now we can combine everything:
\[ \tan 80^\circ \tan 10^\circ + \sin^2 70^\circ + \sin^2 20^\circ = 1 + \cos^2 20^\circ + \sin^2 20^\circ \]

Using the Pythagorean identity \( \sin^2 x + \cos^2 x = 1 \):
\[ \cos^2 20^\circ + \sin^2 20^\circ = 1 \]

Thus, we have:
\[ 1 + 1 = 2 \]

### Final Answer

The value of \( \tan 80^\circ \tan 10^\circ + \sin^2 70^\circ + \sin^2 20^\circ \) is:
\[ \boxed{2} \]