If `sec(alpha-beta)=sqrt(2) and sin(alpha+beta)=(1)/(2)`, then find the least positive values of `alpha and beta`.

Explanation

Given that `sqrt2sin(alpha-beta)=1`
or `sin(alpha-beta)=(1)/(sqrt2)sin45^(@)`
`impliesalpha-beta=45^(@).........................(1)`
Also, `alpha+beta=90^(@)=..............................(2)[:'alpha,beta` are complementary.]
Now, adding (I) and (2) we get,
`2alpha=135^(@)or,alpha=(135^(@))/(2)=67.5^(@)`
From (2) we get, `beta=90^(@)-alpha=90^(@)-67.5^(@)=22.5^(@)`
Hence `alpha=67.5^(@)andbeta=22.5^(@)`.