If Lab Color space covers colors that are not supported by monitors how come photoshop are showing Lab color space in their colorpalette? What is the purpose of it? If I give Lab values as L-(85), a-(29), b-(-99). I am getting a color. But if I use the corresponding RGB values, I am not getting that actual color. Why is that?
There are 3 questions here: What is the purpose of Lab? Why is it in Photoshop? and Why is a color value different in RGB than it is in Lab. There is a ton of litterature on color spaces and Lab is an interesting one.
First, you are right in saying that Lab contains colors that are not supported by monitors, just like your monitor (RGB) can display colors that cannot be printed (CMYK).
Here is a comparison of the main color spaces:
How and why is Lab different from other color spaces?
Basically, color spaces are just different ways to represent colors by using numbers. Lab contains more colors and is more precise because of the 3 axis it uses to represent colors. The L axis represents the luminosity (brightness) of the color, the a axis represents colors from red to green and the b axis represents colors from yellow to blue.
Here is a 3D representation of the Lab color space:
What makes Lab more efficient is that it is independent from other factors. For example, you can use the exact same CMYK recipe on 5 different presses and you'll get 5 different results depending on the paper, the lighting in the room, the ink viscosity and many other factors that required adjustments and compensation by the press operator when the color was printed. Because of that, you cannot figure out a precise CMYK recipe simply by analysing the printed result. The same goes for RGB; for example, some screens use a blueish white, even when they are supposed to be neutral and calibrated. Are the lightbulbs in your office perfectly neutral? Probably not, and the amount of sun you get will also affect the colors you see. This makes RGB and CMYK unreliable when trying to reproduce a specific color.
Lab doesn't take external factors into consideration, therefore it gives you a precise color value by finding the attributes of a color. By giving someone else a Lab color number, they know exactly what color you are refering to.
That explains why you are not getting the same color in RGB. Even if you disregard your monitor and just use the numbers, there is a conversion that happens to fit the Lab color in the smaller RGB space, and that conversion is done according to the RGB color profile that you are using and its settings.
What is the purpose of Lab
There are many things that you can do with Lab. Here are but a few:
It is the best color space to use when it comes to color conversion as it gives you a precise reference point and it offers the widest range of possibilities.
When measuring color with a spectrophotometer you'll usually use Lab, even though most spectrophotometers will also give RGB and CMYK. As I mentionned before, CMYK and RGB are affected by many external factors, which makes them unreliable when you try to reproduce a color.
Because of the 3 axis, a Lab color can easily and clearly be represented on a graph.
In the printing industry, for example in flexographic printing, Lab is used to measure color variations and to adjust the press/ink accordingly.
Finally, why is it in Photoshop?
As you can see above, it is the best tool for color conversion and for getting a precise color reading when trying to precisely reproduce a color. Because of this, many professionals need Lab.