The value of tangent when the angle of right angled triangle equals to $45$ degrees is called $\tan 45^\circ$.

The value of $\tan 45^\circ$ is calculated by calculating the ratio of length of opposite side to length of adjacent side when the angle of the right angled triangle is equal to $45^\circ$.

The value of $\tan 45^\circ$ can be calculated mathematically in two different approaches in mathematics.

1

The value of $\tan 45^\circ$ can be derived in algebraic approach by considering the properties of the triangle when the angle of the right angled triangle is $45^\circ$.

As per properties of the triangle, the lengths of opposite and adjacent sides are exactly equal when the angle of the right angled triangle is $45^\circ$.

For example, $\Delta RPQ$ is a right angled triangle and its angle is $45^\circ$. So, the length of both opposite and adjacent side are same and length of each side is $l$.

$Length \, of \, Opposite \, side$ $=$ $Length \, of \, Adjacent \, side$

$$\implies \frac{Length \, of \, Opposite \, side}{Length \, of \, Adjacent \, side} = 1$$

According to trigonometry, the ratio of length of the opposite side to length of the adjacent side is tangent and the angle of the triangle is $45^\circ$. Therefore, the value of the ratio is known as $\tan 45^\circ$.

$\therefore \, \tan 45^\circ = 1$

2

The value of $\tan 45^\circ$ can also be derived in geometric approach by drawing a line of any length with $45$ degrees angle for constructing a right angled triangle of $45$ degrees angle.

- Firstly, draw a horizontal line, which will be used as base line while using protractor. Call left side point of the horizontal line as Point $L$.
- Use protractor, coincide the centre of the protractor with point $L$ and also coincide horizontal line with right side base line of the protractor. Mark a point at angle $45^\circ$ by considering bottom scale of the protractor.
- Use scale and draw a straight line from point $L$ through $45^\circ$ angle marked point.
- Use compass and set the length to $6.2$ centimetres and draw an arc from point $L$ on $45$ degrees angle line and the arc cuts the $45$ degrees angle line at point $M$.
- Use set square and draw a perpendicular line to horizontal line from $M$ and it intersects the horizontal line at point $N$ perpendicularly.

A right angled triangle ($\Delta MLN$) is formed by this geometric procedure and the angle of this right angled triangle is $45^\circ$.

Use scale and measure the lengths of both opposite side and adjacent side. You will observe that the lengths of both opposite and adjacent sides are equal and the length of each side will be $4.4$ centimetres.

According to definition of tangent in Trigonometry, express it in mathematical form.

$$\tan 45^\circ = \frac{Length \, of \, Opposite \, side}{Length \, of \, Adjacent \, side}$$

$$\implies \tan 45^\circ = \frac{4.4}{4.4}$$

$\therefore \, \tan 45^\circ = 1$

It is proved that tan of angle $45^\circ$ is $1$ in both algebraic and geometric methods.

This value can be expressed in three different forms in mathematics according to three angle measuring systems.

It is usually written in sexagesimal system as,

$\tan 45^° = 1$

It is expressed in circular system as,

$$\tan \Bigg(\frac{\pi}{4}\Bigg) = 1$$

It is expressed in centesimal system as,

$\tan 50^g = 1$