# What is the Half-Angle Identities?

##### 1 Answer

The half-angle identities are defined as follows:

#\mathbf(sin(x/2) = pmsqrt((1-cosx)/2))#

#(+)# for quadrantsIandII

#(-)# for quadrantsIIIandIV

#\mathbf(cos(x/2) = pmsqrt((1+cosx)/2))#

#(+)# for quadrantsIandIV

#(-)# for quadrantsIIandIII

#\mathbf(tan(x/2) = pmsqrt((1-cosx)/(1+cosx)))#

#(+)# for quadrantsIandIII

#(-)# for quadrantsIIandIV

We can derive them from the following identities:

#sin^2(x/2) = (1-cos(x))/2#

#color(blue)(sin(x/2) = pmsqrt((1-cos(x))/2))#

Knowing how **I** and **II** and negative for **III** and **IV**.

#cos^2(x/2) = (1+cos(x))/2#

#color(blue)(cos(x/2) = pmsqrt((1+cos(x))/2))#

Knowing how **I** and **IV** and negative for **II** and **III**.

#color(blue)(tan(x/2) = pmsqrt((1-cos(x))/(1+cos(x))))#

We can see that if we take the conditions for positive and negative values from **I** and *III* and negative for **II** and **IV**.