Prove that tan-1x + cot-1(x+1) = tan-1( 1 + x + x^2)

Lhs=tan-1x+cot-1x=pi/2
=tan-1x+pi/2-tan-1(x+1)
now tan-1x-tan-1y=tan-1{(x-y)/[1-xy]}
so
=tan-1{(x-x-1)/(1+x+x^2)+pi/2
=tan-1{-1/(1+x+x^2)}+pi/2
tan is odd func.
so tan-1(-x)=-tan-1x and tan-1(1/x)=cot-1x
so
-cot-1(1+x+x^2)+pi/2
=tan-1(1+x+x^2)
so
lhs=rhs

We can observe physical properties of elements and compounds without changing the substance.

Examples of physical properties: Density, color, boiling point, state of matter, appearance: dull or shiny, etc. 

But we can also observe and measure chemical properties by reacting a substance with something else. For example, like mixing baking soda and vinegar together. The vinegar reacts with the baking soda and produces carbon dioxide: a new substance. 

Some examples of chemical properties: Flammability, amount of heat that is released during combustion, toxicity (how much damage it causes to other organisms), radioactivity, and ability to oxidize (when you have metal that becomes rusty looking).