8.log4(sin 3π/4)+log9[tan("-" 5π/6)]=　　　　.

解析∵sin3π/4=sin(π"-" π/4)=sinπ/4=√2/2,

　　tan("-" 5π/6)=-tan(π"-" π/6)=tanπ/6=√3/3,

　　∴log4(sin 3π/4)+log9[tan("-" 5π/6)]

　　=log4√2/2+log9√3/3

　　=log_(2^2 ) 2^("-" 1/2)+log_(3^2 ) 3^("-" 1/2)

　　=-1/4-1/4=-1/2.

答案-1/2

9.求下列各式的值:

(1)cos25π/3+tan("-" 15π/4);

(2)sin 810°+tan 765°+tan 1 125°+cos 360°.

解(1)cos25π/3+tan("-" 15π/4)

　　=cos(8π+π/3)+tan("-" 4π+π/4)

　　=cosπ/3+tanπ/4

　　=1/2+1=3/2.

　　(2)原式=sin(2×360°+90°)+tan(2×360°+45°)+tan(3×360°+45°)+cos(0°+360°)=sin 90°+tan 45°+tan 45°+cos 0°=4.

10.导学号93774028设tan(α+8π/7)=a,求(sin(15π/7+α)+3cos(α"-" 13π/7))/(sin(20π/7 "-" α)"-" cos(α+22/7 π) )的值.

解∵tan(α+8π/7)=tan[π+(α+π/7)]

　　=tan(α+π/7)=a,

　　∴原式=(sin(α+π/7)+3cos(α+π/7))/(sin(α+π/7)+cos(α+π/7) )

　　=(tan(α+π/7)+3)/(tan(α+π/7)+1)=(a+3)/(a+1).

11.导学号93774029求证:当k=2或3时,(tan"(" kπ"-" α")·" tan"(" kπ+α")" )/(cos"(" 2kπ"-" α")·" sin"[(" 2k+1")" π+α"]" )=sinα/(cos^3 α).

证明当k=2时,左边=(tan"(" 2π"-" α")·" tan"(" 2π+α")" )/(cos"(" 4π"-" α")·" sin"[(" 4+1")" π+α"]" )=("-" tanα"·" tanα)/(cosα"·" sin"(" π+α")" )=("-" tan^2 α)/("-" cosα"·" sinα)=sinα/(cos^3 α)=右边.

　　当k=3时,左边=(tan"(" 3π"-" α")·" tan"(" 3π+α")" )/(cos"(" 6π"-" α")·" sin"[(" 6+1")" π+α"]" )

　　=(tan"(-" α")·" tanα)/(cos"(-" α")·" sin"(" π+α")" )

　　=("-" tanα"·" tanα)/(cosα"·(-" sinα")" )

　　=sinα/(cos^3 α)=右边.

　　故当k=2或3时,原等式成立.