Statement-1: `sin^(2)6^(@) + sin^(2)12^(@) + sin^(2)18^(@) +……..+ sin^(2)84^(@)=7`
Statement-2: `tan9^(@) tan27^(@) tan45^(@) tan36^(@) tan81^(@)=1`
Statement-3: `(tan(pi/4) + cot(pi/4) + " cosec " pi/4) (tan(pi/4) + cot(pi/4) - " cosec "pi/4) = sec ^(2)pi/3`

Options

A. TTT
B. FFT
C. TTF
D. TFF

Answer

To solve the given statements step by step, we will analyze each statement one by one.

### Statement 1:
**Claim:** \( \sin^2(6^\circ) + \sin^2(12^\circ) + \sin^2(18^\circ) + \ldots + \sin^2(84^\circ) = 7 \)

**Step 1:** Identify the terms in the series.
- The series consists of sine squares from \( 6^\circ \) to \( 84^\circ \) with a common difference of \( 6^\circ \).
- The terms are \( \sin^2(6^\circ), \sin^2(12^\circ), \sin^2(18^\circ), \ldots, \sin^2(84^\circ) \).

**Step 2:** Count the number of terms.
- The angles form an arithmetic sequence where \( a = 6^\circ \), \( d = 6^\circ \), and the last term \( l = 84^\circ \).
- The number of terms \( n \) can be calculated using the formula for the \( n \)-th term of an arithmetic sequence:
\[ n = \frac{l - a}{d} + 1 = \frac{84 - 6}{6} + 1 = 14 \]

**Step 3:** Pair the sine squares.
- Pair the terms: \( \sin^2(6^\circ) + \sin^2(84^\circ) \), \( \sin^2(12^\circ) + \sin^2(78^\circ) \), etc.
- Using the identity \( \sin^2(x) + \sin^2(90^\circ - x) = 1 \):
\[ \sin^2(6^\circ) + \sin^2(84^\circ) = 1 \]
\[ \sin^2(12^\circ) + \sin^2(78^\circ) = 1 \]
\[ \sin^2(18^\circ) + \sin^2(72^\circ) = 1 \]
\[ \sin^2(24^\circ) + \sin^2(66^\circ) = 1 \]
\[ \sin^2(30^\circ) + \sin^2(60^\circ) = 1 \]
\[ \sin^2(36^\circ) + \sin^2(54^\circ) = 1 \]
\[ \sin^2(42^\circ) + \sin^2(48^\circ) = 1 \]
\[ \sin^2(48^\circ) + \sin^2(42^\circ) = 1 \]
\[ \sin^2(54^\circ) + \sin^2(36^\circ) = 1 \]
\[ \sin^2(60^\circ) + \sin^2(30^\circ) = 1 \]
\[ \sin^2(66^\circ) + \sin^2(24^\circ) = 1 \]
\[ \sin^2(72^\circ) + \sin^2(18^\circ) = 1 \]
\[ \sin^2(78^\circ) + \sin^2(12^\circ) = 1 \]
\[ \sin^2(84^\circ) + \sin^2(6^\circ) = 1 \]

- There are 7 pairs, thus:
\[ \text{Total} = 7 \times 1 = 7 \]

**Conclusion for Statement 1:** True.

### Statement 2:
**Claim:** \( \tan(9^\circ) \tan(27^\circ) \tan(45^\circ) \tan(36^\circ) \tan(81^\circ) = 1 \)

**Step 1:** Simplify the terms.
- Note that \( \tan(45^\circ) = 1 \).
- Rewrite the product:
\[ \tan(9^\circ) \tan(27^\circ) \tan(36^\circ) \tan(81^\circ) \]

**Step 2:** Use the identity \( \tan(90^\circ - x) = \cot(x) \).
- Thus, \( \tan(81^\circ) = \cot(9^\circ) \) and \( \tan(36^\circ) = \cot(54^\circ) \).
- Therefore:
\[ \tan(9^\circ) \tan(81^\circ) = 1 \]
\[ \tan(27^\circ) \tan(63^\circ) = 1 \]

**Step 3:** Check if the product equals 1.
- The product simplifies to:
\[ 1 \cdot \tan(27^\circ) \cdot \tan(36^\circ) \cdot 1 \]
- Since \( \tan(27^\circ) \tan(63^\circ) = 1 \) and \( \tan(36^\circ) \tan(54^\circ) = 1 \), the entire product does not equal 1.

**Conclusion for Statement 2:** False.

### Statement 3:
**Claim:** \( ( \tan(\frac{\pi}{4}) + \cot(\frac{\pi}{4}) + \csc(\frac{\pi}{4}) ) ( \tan(\frac{\pi}{4}) + \cot(\frac{\pi}{4}) - \csc(\frac{\pi}{4}) ) = \sec^2(\frac{\pi}{3}) \)

**Step 1:** Calculate the left-hand side.
- \( \tan(\frac{\pi}{4}) = 1 \), \( \cot(\frac{\pi}{4}) = 1 \), \( \csc(\frac{\pi}{4}) = \sqrt{2} \).
- Substitute these values:
\[ (1 + 1 + \sqrt{2})(1 + 1 - \sqrt{2}) = (2 + \sqrt{2})(2 - \sqrt{2}) \]

**Step 2:** Expand the product.
- Using the difference of squares:
\[ (2 + \sqrt{2})(2 - \sqrt{2}) = 4 - 2 = 2 \]

**Step 3:** Calculate the right-hand side.
- \( \sec^2(\frac{\pi}{3}) = 1 + \tan^2(\frac{\pi}{3}) = 1 + 3 = 4 \).

**Conclusion for Statement 3:** False.

### Final Conclusions:
- Statement 1: True
- Statement 2: False
- Statement 3: False