Question

What are some important formulae of trigonometry?

Answer 1

There are many ways to work on trigonometric problems. But for that you must be very clear about all the trignometric identities and trigonometric formulae first.
Important Properties and Formulas
Basic Identities
$sinx=\frac{1}{csc{x}^{\prime }}tanx=\frac{sinx}{cos{x}^{\prime }}cosx=\frac{1}{sec{x}^{\prime }}cotx=\frac{cosx}{sinx}tanx=\frac{1}{cotx}sin(-x)=-sinx,cos(-x)=-cosx,tan(-x)=-tanx$

Pythagorean Identities
$si{n}^{2}x+co{s}^{2}x=1,1+co{t}^{2}x=cs{c}^{2}x,1+ta{n}^{2}x=se{c}^{2}x$

Sum and Difference Identities
$sin(u\pm v)=sinucosv\pm cosusinv,cos(u\pm v)=cosucosv\mp sinusinv,tan(u\pm v)=\frac{tanu\pm tanv}{1\mp tanutanv}$

Double-Angle~Identities
$sin2x=2sinxcosx,cos2x=co{s}^{2}x-si{n}^{2}x=1-2si{n}^{2}x=2co{s}^{2}x-1,tan2x=\frac{2tanx}{1-ta{n}^{2}x}$

Half- Angle~Identities
$sin\frac{x}{2}=\pm \sqrt{\frac{1-cosx}{2}},cos\frac{x}{2}=\pm \sqrt{\frac{1+cosx}{2}}tan\frac{x}{2}=\pm \sqrt{\frac{1-cosx}{1+cosx}}=\frac{sinx}{1+cosx}=\frac{1-cosx}{sinx}$