特殊三角函数值化简

题目

特殊三角函数值化简
a^3 tan45°+a^2 btan60°+9ab^2 cot^2 60°+2b^3cos60°
2cos^2 45°-3a cot45°+a^2tan^2 60°-2a^3 sin^2 45°
4a^2 sin^4 45°-6ab tan^2 30°+（b*cot45°）^2

答案

a^3 tan⁡〖45°〗+a^2 b tan⁡〖60°〗+9ab^2 cot^2⁡〖60°〗+2b^3 cos⁡〖60°〗
=a^3 (1)+a^2 b√(3)+9ab^2 (1/3)+2b^3 (1/2)
=a^3+〖√(3) a〗^2 b+3ab^2+b^3
2 cos^2⁡〖45°〗-3a cot⁡〖45°〗+a^2 tan^2⁡〖60°〗-2a^3 sin^2⁡〖45°〗
=2(1/2)-3a(1)+a^2 (3)-2a^3 (1/2)
=1-3a+3a^2-a^3
=-(a-1)^3
4a^2 sin^4⁡〖45°〗-6ab tan^2⁡〖30°〗+(b cot⁡〖45°〗 )^2
=4a^2 (1/4)-6ab(1/3)+(b(1) )^2
=a^2-2ab+b^2
=(a-b)^2