LHS=1/(cos290^@)+1/(sqrt3sin250^@)
=1/(cos(360-70)^@)+1/(sqrt3sin(180+70)^@)
=1/(cos70^@)-1/(sqrt3sin70^@)
=(sqrt3sin70^@-cos70^@)/(sqrt3sin70^@cos70^@)
=1/sqrt3[(2{sqrt3sin70^@-cos70^@})/(2sin70^@cos70^@)]
=1/sqrt3[(2*2{sin70^@*(sqrt3/2)-cos70^@*(1/2)})/(sin140^@)]
=1/sqrt3[(4{sin70^@*cos30^@-cos70^@*sin30^@})/(sin(180-40)^@)]
=1/sqrt3[(4{sin(70-30)^@})/(sin40^@)]=1/sqrt3[(4{cancel(sin40^@)})/cancel((sin40^@))]=4/sqrt3=RHS
NOTE that cos(360-A)^@=cosA and sin(180+A)^@=-sinA