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求证cos(a+b)*cos(a-b)=cos^2a-sin^2b求详解
题目详情
求证cos(a+b)*cos(a-b)=cos^2 a-sin^2 b 求详解
▼优质解答
答案和解析
cos(a+b)=cosacosb-sinasinb
cos（a-b）=cosacosb+sinasinb
所以
cos(a+b)*cos(a-b)=cos^2acos^2b-sin^2asin^2b
=cos^2a(1-sin^2b)-sin^2asin^2b
=cos^2a-cos^2asin^2b-sin^2sin^2b
=cos^2a-sin^2b(cos^2a+sin^2a)
=cos^2a-sin^2b
cos（a-b）=cosacosb+sinasinb
所以
cos(a+b)*cos(a-b)=cos^2acos^2b-sin^2asin^2b
=cos^2a(1-sin^2b)-sin^2asin^2b
=cos^2a-cos^2asin^2b-sin^2sin^2b
=cos^2a-sin^2b(cos^2a+sin^2a)
=cos^2a-sin^2b
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