已知tana= 3 计算(5cos^2a-3sin^2a)/(1+sin^2a)

答：
tana=3,sina=3cosa
代入sin²a+cos²a=1得：cos²a=1/10
(5cos²a-3sin²a) /(1+sin²a)
=(5-3tan²a) /(1/cos²a+tan²a)
=(5-27) / (10+9)
=-22/19

(5cos²a-3sin²a)/(1+sin²a)

=(5cos²a-3sin²a)/(sin²a+cos²a+sin²a)

=(5-3tan²a)/(2tan²a+1) （分子分母同时除以cos²a而得）

=(5-3×3²)/(2×3²+1)

=（5-27）/(18+1)

=-22/19

原式
=(5cos^2a-3sin^2a)/(sin^2a+cos^2a+sin^2a)
=(5cos^2a-3sin^2a)/(2sin^2a+cos^2a)
上下同除cos^2a
=(5-3tan^2a)/(2tan^2a+1)
=(5-3*3^2)/(2*3^2+1)
=(5-3*9)/(2*9+1)
=-22/19
...

(5cos^2a-3sin^2a)/(1+sin^2a)
=(5cos^2a-3sin^2a)/(sin^2a+cos^2a+sin^2a)
=(5cos^2a-3sin^2a)/(2sin^2a+cos^2a)(分子分母同时除以cos^2a）
=(5cos^2a/cos^2a-3sin^2a/cos^2a)/(2sin^2a/cos^2a+cos^2a/cos^2a)
=(5-3tan^2a)/(2tan^2a+1)
=(5-3*3^2)/(2*3^2+1)
=(5-27)/(18+1)
=-22/19