# Identity Index in Trigonometry

Here is the formula index in trigonometric identities. Again, identities are the fundamentals of trigonometry. This is the counterpart of four basic operations in arithmetic.

Basic Identity

$sinx=\displaystyle\frac{1}{cscx}$

$cosx=\displaystyle\frac{1}{secx}$

$tanx=\displaystyle\frac{1}{cotx}$

Pythagorean Identity

$cos^2x+sin^2x=1$

$sec^2x=1+tan^2x$

$csc^2x=1+cot^2x$

$cos(x+y)=cosxcosy-sinxsiny$

$cos(x-y)=cosxcosy+sinxsiny$

$sin(x+y)=sinxcosy+cosxsiny$

$sin(x-y)=sinxcosy-cosxsiny$

$tan(x+y)=\displaystyle\frac{tanx+tany}{1-tanxtany}$

$tan(x-y)=\displaystyle\frac{tanx-tany}{1+tanxtany}$

$cot(x+y)= \displaystyle\frac{cotxcoty-1}{cotx+coty}$

$cot(x-y)= \displaystyle\frac{cotxcoty+1}{cotx-coty}$

Special Reduction Identity

$sin(-x)=-sinx$

$cos(-x)=cosx$

$tan(-x)=-tanx$

$cot(-x)=-cotx$

$cos(\frac{\pi}{2}-x)=cosx$

$sin(\frac{\pi}{2}-x)=sinx$

$tan(\frac{\pi}{2}-x)=tanx$

$cot(\frac{\pi}{2}-x)=cotx$

Double Angle Identity

$sin2x=2sinxcosx$

$cos2x=cos^2x-sin^2x=1-2sin^2x=2sin^2x-1$

$tan2x=\displaystyle\frac{2tanx}{1-tan^2x}$

Half Angle Identity

$sin(\displaystyle\frac{x}{2})=\pm\sqrt{\displaystyle\frac{1-cosx}{2}}$

$cos(\displaystyle\frac{x}{2})=\pm\sqrt{\displaystyle\frac{1+cosx}{2}}$

$tan(\displaystyle\frac{x}{2})=\displaystyle\frac{sinx}{1+cosx}=\pm\sqrt{\displaystyle\frac{1-cosx}{1+cosx}}$

Product to Sum Identity

$sinxcosy=\displaystyle\frac{1}{2}[sin(x+y)+sin(x-y)]$

$cosxsiny=\displaystyle\frac{1}{2}[sin(x+y)-sin(x-y)]$

$cosxcosy=\displaystyle\frac{1}{2}[cos(x+y)+cos(x-y)]$

$sinxsiny=\displaystyle\frac{1}{2}[cos(x-y)-sin(x+y)]$

Sum to Product Identity

$sinx+siny=2sin\displaystyle\frac{x+y}{2}\cdot cos\displaystyle\frac{x-y}{2}$

$sinx-siny=2cos\displaystyle\frac{x+y}{2}\cdot sin\displaystyle\frac{x-y}{2}$

$cosx+cosy=2cos\displaystyle\frac{x+y}{2}\cdot cos\displaystyle\frac{x-y}{2}$

$cosx-cosy=-2sin\displaystyle\frac{x+y}{2}\cdot sin\displaystyle\frac{x-y}{2}$

If you want to add more, feel free to add in comment box and I’m glad to include that.