Find the value of `45^@` of all trigonometrically ratios geometrically. and evaluate the following . `(1- tan^(2)45^@)/(1+tan^(2)45^@)`

Answer

To solve the problem step by step, we will first find the trigonometric ratios for \(45^\circ\) and then evaluate the expression \(\frac{1 - \tan^2 45^\circ}{1 + \tan^2 45^\circ}\).

### Step 1: Construct a Right-Angle Triangle
We start by constructing a right-angle triangle where one of the angles is \(45^\circ\). In a \(45^\circ-45^\circ-90^\circ\) triangle, the two legs are equal.

Let the lengths of both legs be \(a\).

### Step 2: Find the Hypotenuse
Using the Pythagorean theorem, we can find the length of the hypotenuse \(BC\):
\[ BC = \sqrt{AB^2 + AC^2} = \sqrt{a^2 + a^2} = \sqrt{2a^2} = a\sqrt{2} \]

### Step 3: Calculate Trigonometric Ratios
Now, we can find the trigonometric ratios for \(45^\circ\):

1. **Sine**:
\[ \sin 45^\circ = \frac{\text{Opposite}}{\text{Hypotenuse}} = \frac{a}{a\sqrt{2}} = \frac{1}{\sqrt{2}} \]

2. **Cosine**:
\[ \cos 45^\circ = \frac{\text{Adjacent}}{\text{Hypotenuse}} = \frac{a}{a\sqrt{2}} = \frac{1}{\sqrt{2}} \]

3. **Tangent**:
\[ \tan 45^\circ = \frac{\text{Opposite}}{\text{Adjacent}} = \frac{a}{a} = 1 \]

### Step 4: Evaluate the Expression
Now we need to evaluate the expression:
\[ \frac{1 - \tan^2 45^\circ}{1 + \tan^2 45^\circ} \]
Substituting \(\tan 45^\circ = 1\):
\[ = \frac{1 - 1^2}{1 + 1^2} = \frac{1 - 1}{1 + 1} = \frac{0}{2} = 0 \]

### Final Answer
The value of the expression is \(0\).

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