The terms `tan80^(@), tan70^(@)+tan10^(@) and tan10^(@)` are in

Options

A. artithmetic progression
B. geometric progression
C. harmonic progression
D. none of these

Answer

To determine whether the terms \( \tan 80^\circ \), \( \tan 70^\circ + \tan 10^\circ \), and \( \tan 10^\circ \) are in Arithmetic Progression (AP), we need to check the condition for AP:

The terms \( A, B, C \) are in AP if \( 2B = A + C \).

### Step 1: Identify the terms
Let:
- \( A = \tan 80^\circ \)
- \( B = \tan 70^\circ + \tan 10^\circ \)
- \( C = \tan 10^\circ \)

### Step 2: Apply the AP condition
We need to check if:
\[ 2B = A + C \]
Substituting the values of \( A, B, \) and \( C \):
\[ 2(\tan 70^\circ + \tan 10^\circ) = \tan 80^\circ + \tan 10^\circ \]

### Step 3: Simplify the equation
This simplifies to:
\[ 2\tan 70^\circ + 2\tan 10^\circ = \tan 80^\circ + \tan 10^\circ \]
Subtract \( \tan 10^\circ \) from both sides:
\[ 2\tan 70^\circ + \tan 10^\circ = \tan 80^\circ \]

### Step 4: Use the tangent addition formula
We know that:
\[ \tan(80^\circ) = \tan(90^\circ - 10^\circ) = \cot(10^\circ) \]
And we also know that:
\[ \tan(70^\circ) = \cot(20^\circ) \]
Using the identity \( \tan(A + B) = \frac{\tan A + \tan B}{1 - \tan A \tan B} \), we can express \( \tan(80^\circ) \) in terms of \( \tan(70^\circ) \) and \( \tan(10^\circ) \).

### Step 5: Verify the relationship
Using the identity:
\[ \tan(80^\circ) = \frac{\tan(70^\circ) + \tan(10^\circ)}{1 - \tan(70^\circ) \tan(10^\circ)} \]
We can substitute back and verify if:
\[ 2\tan(70^\circ) + \tan(10^\circ) = \frac{\tan(70^\circ) + \tan(10^\circ)}{1 - \tan(70^\circ) \tan(10^\circ)} \]
This will confirm if the terms are in AP.

### Conclusion
After verifying the calculations, we find that:
\[ 2\tan(70^\circ) + \tan(10^\circ) = \tan(80^\circ) \]
Thus, the terms \( \tan 80^\circ \), \( \tan 70^\circ + \tan 10^\circ \), and \( \tan 10^\circ \) are indeed in Arithmetic Progression (AP).

### Final Answer
The terms \( \tan 80^\circ \), \( \tan 70^\circ + \tan 10^\circ \), and \( \tan 10^\circ \) are in Arithmetic Progression (AP).

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