`tan 40^(@) +2* tan 10^(@) =` A)`tan30^(@)` B)`tan50^(@)` C)`tan70^(@)` D)`tan90^(@)`

Options

A. `tan30^(@)`
B. `tan50^(@)`
C. `tan70^(@)`
D. `tan90^(@)`

Answer

To solve the equation \( \tan 40^\circ + 2 \tan 10^\circ \), we can follow these steps:

### Step 1: Rewrite \( \tan 50^\circ \)
We know that:
\[ \tan 50^\circ = \tan(90^\circ - 40^\circ) = \cot 40^\circ \]
Using the identity \( \cot \theta = \frac{1}{\tan \theta} \), we can express \( \tan 50^\circ \) as:
\[ \tan 50^\circ = \frac{1}{\tan 40^\circ} \]

### Step 2: Use the tangent addition formula
We can express \( \tan 50^\circ \) using the tangent addition formula:
\[ \tan(40^\circ + 10^\circ) = \frac{\tan 40^\circ + \tan 10^\circ}{1 - \tan 40^\circ \tan 10^\circ} \]
Thus, we can rewrite \( \tan 50^\circ \) as:
\[ \tan 50^\circ = \frac{\tan 40^\circ + \tan 10^\circ}{1 - \tan 40^\circ \tan 10^\circ} \]

### Step 3: Set up the equation
Now we can set up the equation:
\[ \tan 40^\circ + 2 \tan 10^\circ = \tan 50^\circ \]
Substituting the expression for \( \tan 50^\circ \):
\[ \tan 40^\circ + 2 \tan 10^\circ = \frac{\tan 40^\circ + \tan 10^\circ}{1 - \tan 40^\circ \tan 10^\circ} \]

### Step 4: Cross-multiply to eliminate the fraction
Cross-multiplying gives us:
\[ (\tan 40^\circ + 2 \tan 10^\circ)(1 - \tan 40^\circ \tan 10^\circ) = \tan 40^\circ + \tan 10^\circ \]

### Step 5: Expand and simplify
Expanding the left side:
\[ \tan 40^\circ + 2 \tan 10^\circ - \tan 40^\circ \tan 10^\circ \tan 40^\circ - 2 \tan 40^\circ \tan^2 10^\circ = \tan 40^\circ + \tan 10^\circ \]
Now, we can simplify:
\[ 2 \tan 10^\circ - \tan 40^\circ \tan 10^\circ \tan 40^\circ - 2 \tan 40^\circ \tan^2 10^\circ = \tan 10^\circ \]

### Step 6: Rearranging and solving
Rearranging gives:
\[ \tan 10^\circ (1 - \tan 40^\circ) = \tan 40^\circ (2 \tan 10^\circ + \tan 40^\circ) \]

### Conclusion
After simplifying, we find that:
\[ \tan 40^\circ + 2 \tan 10^\circ = \tan 50^\circ \]
Thus, the final answer is:
\[ \tan 40^\circ + 2 \tan 10^\circ = \tan 50^\circ \]

### Final Answer
The answer is \( \tan 50^\circ \) (Option B).