If sin α=45 and cos β=513, prove that cosα−β2=8√65
We have,
sin α=45 and cos β=513⇒ cos α=35 and sin β=1213∴ cos(α−β)=cos α cos β+sin α.sin β=35.513+45.1213=1565+4865=6365
Now,
cos(α−β2)=√1+cos(α−β)2=√1+63652=√12865×2=√6465=±8√65∴ cos (α−β2)=8√65