(1)
-⅔+(4/3)cos²15°
=-⅔+⅔(1+cos30°)
=-⅔+⅔+⅔·(√3/2)
=√3/3
(2)
π≤4≤3π/2
sin4<0,cos4<0
2√(sin8+1)+√(2cos8+2)
=2√(2sin4cos4+sin²4+cos²4)+√[2(1+cos8)]
=2√(sin4+cos4)²+√(2·2cos²4)
=2|sin4+cos4|+2|cos4|
=-2(sin4+cos4)-2cos4
=-2sin4-4cos4
(3)
tanα=1/7<√3/3
α为锐角,0<α<π/6
tanβ=1/3,1/3<√3/3
β为锐角,0<β<π/6
0<α+2β<π/2,α+2β为锐角
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-tanαtan²β)/(1-2tanαtanβ-tan²β)
=[1/7 +2·(1/3)-(1/7)(1/3)²]/[1-2·(1/7)(1/3)-(1/3)²]
=1
α+2β=π/4