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

Addition and Subtraction Identify

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.

 

 

 

 

 

 

 

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