Tg^2a+ctg^2a-6/tg^2a+ctg^2a+2

Tg^2a+ctg^2a-6=sin²a/cos²a+cos²a/sin²a - 6=

= (1-cos2a) / (1+cos2a) + (1+cos2a) / (1-cos2a) - 6=

= (1-2cos2a+cos²2a+1+2cos2a+cos²2a - 6+6cos²2a) / (1-cos²2a) =

= (8cos²2a-4) / (1-cos²2a) = 4 (2cos²2a-1) / sin²2a=4cos4a/sin²2a

Tg^2a+ctg^2a + 2=sin²a/cos²a+cos²a/sin²a + 2=

= (1-cos2a) / (1+cos2a) + (1+cos2a) / (1-cos2a) + 2=

= (1-2cos2a+cos²2a+1+2cos2a+cos²2a + 2 - 2cos²2a) / (1-cos²2a) =

=4 / (1-cos²2a) = 4/sin²2a

(Tg^2a+ctg^2a - 6) / (Tg^2a+ctg^2a + 2) = 4cos4a/sin²a:4/sin²a=

=4cos4a/sin²2a * sin²2a/4=cos4a