Question

Write down all trigonometric identies.

Answer 1

Pythagorean Identities:
$si{n}^2\theta +co{s}^2\theta =1ta{n}^2\theta +1=se{c}^2\theta co{t}^2\theta +1=cs{c}^2\theta$

Reciprocal Identities:
$sinx=\frac{1}{cscx}cscx=\frac{1}{sinx}cosx=\frac{1}{secx}secx=\frac{1}{cosx}tanx=\frac{1}{cotx}cotx=\frac{1}{tanx}$

Odd and Even Identities
$sin(-x)=-sinxcsc(-x)=-cscxcos(-x)=cosxsec(-x)=secxtan(-x)=-tanxcot(-x)=-cotx$

Sum and Difference Identities:
$sinx=(x+y)=sinxcosy+cosxsinysinx=(x+y)=sinxcosy+cosxsinycos(x+y)=cosxcosy-sinxsiny-sinxsinycos(x-y)=cosxcosy-sinxsiny-sinxsinytan(x+y)=\frac{tanx+tany}{1-tanxtany}tan(x-y)=\frac{tanx-tany}{1+tanxtany}$

Double Angle Identities:
$sin2x=2sinxcosxcos2x=\{\begin{array}{c}co{s}^{2}x-si{n}^{2}x\\ 1-2si{n}^{2}x\\ 2co{s}^{2}x-1\end{array}tan2x=\frac{2tanx}{1-ta{n}^{2}x}$

Half Angle Identities:
$sin\frac{x}{2}=\pm \sqrt{\frac{1-cosx}{2}}orsi{n}^{2}x=\frac{1-cos2x}{2}cos\frac{x}{2}=\pm \sqrt{\frac{1-cosx}{2}}orco{s}^{2}x=\frac{1+cos2x}{2}tan\frac{x}{2}=\pm \sqrt{\frac{1-cosx}{1+cosx}}or\frac{sinx}{1+cosx}or\frac{1-cosx}{sinx}$

Product to Sum Identities:
$cosxcosy=\frac{1}{2}\left[cos\right(x+y)+cos(x-y\left)\right]sinxsiny=\frac{1}{2}\left[cos\right(x-y)-cos(x+y\left)\right]sinxcosy=\frac{1}{2}\left[sin\right(x+y)+sin(x-y\left)\right]$

Sum to product Identities:
$sinx+siny=2sin\left(\frac{x+y}{2}\right)cos\left(\frac{x-y}{2}\right)sinx-siny=2cos\left(\frac{x+y}{2}\right)sin\left(\frac{x-y}{2}\right)cosx+cosy=2cos\left(\frac{x+y}{2}\right)cos\left(\frac{x-y}{2}\right)cosx-cosy=-2sin\left(\frac{x+y}{2}\right)sin\left(\frac{x-y}{2}\right)$