1)
Т.к. п/2 < x < п, поэтому:
sin(a)>0, cos(a)<0, tg(a)<0.<br>
ctg(a) = cos(a) / sin(a) = 1 / tg(a)
tg(a) = 1 / ctg(a) = - 1/sqr(2) = - sqr(2)/2
ctg(a) = cos(a) / sin(a)
ctg(a)^2 = cos(a)^2 / sin(a)^2
ctg(a)^2 = [ 1 - sin(a)^2 ] / sin(a)^2
sin(a)^2 ctg(a)^2 = 1 - sin(a)^2
sin(a)^2 [ 1 + ctg(a)^2 ] = 1
sin(a)^2 = 1 / [ 1 + ctg(a)^2 ]
sin(a) = 1 / sqr( 1 + ctg(a)^2 )
sin(a) = 1 / sqr( 1 + 2) = 1/sqr(3) = sqr(3)/3
cos(a) = - sqr(1 - sin(a)^2)
cos(a) = - sqr(1 - 1/3) = - sqr(2/3)
2)
sin(2a)^2 + cos(2a)^2 + ctg(5a)^2 = 1 + ctg(5a)^2 =
= 1 + cos(5a)^2 / sin(5a)^2 = [ sin(5a)^2 + cos(5a)^2 ] / sin(5a)^2 =
= 1 / sin(5a)^2
ctg(a) / [ tg(a) + ctg(a) ] = cos(a) / [sin(a) ( sin(a)/cos(a) + cos(a)/sin(a) ) ] =
= cos(a)^2 / [ sin(a)^2 + cos(a)^2 ] = cos(a)^2