化简tan111+tan114+tan111tan114还有这个sin(x27)cos(18-x)-sin(63-x)sin(x-18)以及(sin7+cos15sin8)/(cos7-sin15sin8)

Question

化简tan111+tan114+tan111tan114
还有这个sin(x 27)cos(18-x)-sin(63-x)sin(x-18)
以及(sin7+cos15sin8)/(cos7-sin15sin8)

Answer 1

（1） 1=tan45°=tan225°=tan(111°+114°)
=(tan111°+tan114°)/(1-tan111°tan114°)
则 1-tan111°tan114°= tan111°+tan114°,
得 tan111°+tan114°+ tan111°tan114°=1.
（2）sin(x+ 27°)cos(18°-x) - sin(63°-x)sin(x-18°)
=cos[90°-(x+ 27°)]cos(18°-x) - sin(63°-x)sin(x-18°)
=cos(63°-x)cos(18°-x) - sin(63°-x)sin(x-18°)
= cos[63°-x)+(x-18°)] = cos45°=√2/2.
(3) (sin7°+cos15°sin8°)/(cos7°-sin15°sin8°)
= (2sin7°+2cos15°sin8°)/(2cos7°-2sin15°sin8°)
= (2sin7°+sin23°-sin7°)/(2cos7°+cos23°-cos7°)
= (sin7°+sin23°)/(cos7°+cos23°)
= 2sin15°cos8°/(2cos15°cos8°)
=tan15°= sin30°/(1+cos30°) = 1/(√3+1) = (√3-1)/2.