Question

证明三角恒等式2sin4x+

3

4

sin22x+5cos4x-cos3xcosx=2(1+cos2x)．

Answer 1

证明：左边=2sin⁴x+

3

4

（2sinxcosx）²+5cos⁴x-cos（2x+x）cosx
=2sin⁴x+3sin²xcos²x+5cos⁴x-（cos2xcosx-sin2xsinx）cosx
=2sin⁴x+3sin²xcos²x+5cos⁴x-[（2cos²x-1）cosx-2sin²xcosx]cosx
=2sin⁴x+3sin²xcos²x+5cos⁴x-[2cos³x-cosx-2（1-cos²x）cosx]cosx
=2sin⁴x+3sin²xcos²x+5cos⁴x-（4cos³x-3cosx）cosx
=2sin⁴x+3sin²xcos²x+cos⁴x+3cos²x
=（2sin²x+cos²x）（sin²x+cos²x）+3cos²x
=2sin²x+cos²x+3cos²x
=2+2cos²x=2（1+cos²x）=右边