Question

Without using mathematical tables or calculator, find, in surd form(radicals) the value of tan22.5​

Answer 1

Explanation:

tan 45=1

`tan 2\alpha =(2tan\alpha)/(1-tan^2\alpha ) \\put~\alpha =22.5\\2\alpha =2 * 22.5=45\\tan~2\alpha =tan~45=1\\1=(2tan \alpha )/(1-tan^2\alpha) \\1-tan^2\alpha =2 tan\alpha \\ tan^2\alpha +2tan \alpha -1=0\\tan\alpha =(-2 \pm√(4+4) )/(2) \\=-1\pm√(2) \\\alpha =22.5\\so ~tan~\alpha ~is~positive.\\so ~tan~\alpha =√(2) -1`