Упростить: 1) tg^2a + sin^2a - 1/cos^2a 2) (cos3a + cosa) / 2cosa + 2sin^2a

Воспользуемся следующими формулами из тригонометрии:

sin^2 (α) + cos^2 (α) = 1,

cos (2 * α) = 1 - 2 * sin^2 (α),

cos (α) + cos (β) = 2 * cos[ (α + β) / 2] * cos[ (α - β) / 2;

1)

tg^2 (α) + sin^2 (α) - 1/cos^2 (α) =

= sin^2 (α) / cos^2 (α) - 1/cos^2 (α) + sin^2 (α) =

= [sin^2 (α) - 1]/cos^2 (α) + sin^2 (α) =

= - cos^2 (α) / cos^2 (α) + sin^2 (α) =

= - 1 + sin^2 (α) = - cos^2 (α).

2)

[cos (3 * α) + cos (α) ]/[2 * cos (α) ] + 2 * sin^2 (α) =

= 2 * cos (2 * α) * cos (α) / [2 * cos (α) ] + 2 * sin^2 (α) =

= cos (2 * α) + 2 * sin^2 (α) =

= 1 - 2 * sin^2 (α) + 2 * sin^2 (α) = 1.