-? +(4/3)cos？ 15

=-? +? ( 1+cos30)

=-? +? +? (√3/2)

=√3/3

(2)

π≤4≤3π/2

sin4 & lt0，cos4 & lt0

2√(sin8+ 1)+√(2cos8+2)

=2√(2sin4cos4+sin？ 4+cos？ 4)+√[2( 1+cos8)]

=2√(sin4+cos4)？ +√(2 2co? 4)

=2|sin4+cos4|+2|cos4|

=-2(sin4+cos4)-2cos4

=-2sin4-4cos4

(3)

tanα= 1/7 & lt； √3/3

α is an acute angle, 0.

tanβ= 1/3， 1/3 & lt； √3/3

β is an acute angle, 0.

0 & ltα+2β& lt； π/2 and α+2β are acute angles.

tan(α+2β)=[tan(α+β)+tanβ]/[ 1-tan(α+β)tanβ]

=[(tanα+tanβ)/( 1-tanαtanβ)+tanβ]/( 1-(tanα+tanβ)tanβ/( 1-tanαtanβ)]

=[tanα+tanβ+tanβ( 1-tanαtanβ)]/[ 1-tanαtanβ-(tanα+tanβ)tanβ]

=(tanα+2tanβ-tanαtan？ β)/( 1-2tanαtanβ-tan？ β)

=[ 1/7 +2 ( 1/3)-( 1/7)( 1/3)? ]/[ 1-2 ( 1/7)( 1/3)-( 1/3)? ]

= 1

α+2β=π/4