What is the value of `( tan^(2) 25^(@) )/( cosec^(2) 65^(@) ) + ( cot^(2) 25^(@) ) /( sec^(2) 65^(@) ) + 2 tan 20^(@) tan 45^(@) tan 70^(@)?`

Options

A. 1
B. 2
C. 3
D. 4

Answer

To solve the given problem step by step, we will use trigonometric identities and properties.

### Step 1: Rewrite the Expression
The expression we need to evaluate is:
\[ \frac{\tan^2 25^\circ}{\csc^2 65^\circ} + \frac{\cot^2 25^\circ}{\sec^2 65^\circ} + 2 \tan 20^\circ \tan 45^\circ \tan 70^\circ \]

### Step 2: Use Trigonometric Identities
We know that:
- \(\csc^2 \theta = \sec^2 (90^\circ - \theta)\)
- \(\sec^2 \theta = \csc^2 (90^\circ - \theta)\)

Using these identities, we can rewrite:
- \(\csc^2 65^\circ = \sec^2 25^\circ\) (since \(65^\circ = 90^\circ - 25^\circ\))
- \(\sec^2 65^\circ = \csc^2 25^\circ\)

### Step 3: Substitute into the Expression
Now substituting these into the expression:
\[ \frac{\tan^2 25^\circ}{\sec^2 25^\circ} + \frac{\cot^2 25^\circ}{\csc^2 25^\circ} + 2 \tan 20^\circ \tan 45^\circ \tan 70^\circ \]

### Step 4: Simplify Each Term
1. For the first term:
\[ \frac{\tan^2 25^\circ}{\sec^2 25^\circ} = \tan^2 25^\circ \cdot \cos^2 25^\circ = \sin^2 25^\circ \]

2. For the second term:
\[ \frac{\cot^2 25^\circ}{\csc^2 25^\circ} = \cot^2 25^\circ \cdot \sin^2 25^\circ = \cos^2 25^\circ \]

3. The third term simplifies as follows:
\[ \tan 45^\circ = 1 \Rightarrow 2 \tan 20^\circ \cdot 1 \cdot \tan 70^\circ = 2 \tan 20^\circ \tan 70^\circ \]
Using the identity \(\tan(90^\circ - \theta) = \cot \theta\), we have:
\[ \tan 70^\circ = \cot 20^\circ \Rightarrow 2 \tan 20^\circ \tan 70^\circ = 2 \tan 20^\circ \cdot \cot 20^\circ = 2 \]

### Step 5: Combine the Results
Now we can combine all the results:
\[ \sin^2 25^\circ + \cos^2 25^\circ + 2 \]

Using the Pythagorean identity:
\[ \sin^2 \theta + \cos^2 \theta = 1 \]
Thus:
\[ 1 + 2 = 3 \]

### Final Answer
The final value of the expression is:
\[ \boxed{3} \]