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Giải bài 67 trang 106 SBT toán 10 - Cánh diều

Cho góc nhọn \(\alpha \). Biểu thức (sin\(\alpha \). cot\(\alpha \))2 + (cos\(\alpha \) . tan\(\alpha \))2 bằng:

T🃏ổng hợp đề thi 👍học kì 2 lớp 10 tất cả các môn - Cánh diều

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Đề bài

Cho góc nhọn \(\alpha \). Biểu thức (sin\(\alpha \). cot\(\alpha \))2 + (cos\(\alpha \) . tan\(\alpha \))2 bằng:

A. 2     B. tan²\(\alpha \) + cot²\(\alpha \)      C. 1     D. sin\(\alpha \) + cos\(\alpha \)

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Lời giải chi tiết

Ta có: \(\tan \alpha  = \frac{{\sin \alpha }}{{\cos \alpha }},\cot \alpha  = \frac{{\cos \alpha }}{{\sin \alpha }}\)\( \Rightarrow {\left( {\sin \alpha .\cot \alpha } \right)^2} + {\left( {\cos \alpha .\tan \alpha } \right)^2} = {\left( {\sin \alpha .\frac{{\cos \alpha }}{{\sin \alpha }}} \right)^2} + {\left( {\cos \alpha .\frac{{\sin \alpha }}{{\cos \alpha }}} \right)^2}\)\( = {\cos ^2}\alpha  + {\sin ^2}\alpha  = 1\)

Chọn C

 

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