Toán Lớp 9: Chứng minh rằng 1 + tan² a = 1/cos² a , 1+ cot² a = 1/ sin² a

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   \(a)1+\tan^2\alpha=\dfrac{1}{\cos^2\alpha}\\\text{Xét VP =}\dfrac{1}{\cos^2\alpha}\\\quad\quad\quad=\dfrac{\sin^2\alpha+\cos^2\alpha}{\cos^2\alpha}\\\quad\quad\quad=\dfrac{\sin^2\alpha}{\cos^2\alpha}+\dfrac{\cos^2\alpha}{\cos^2\alpha}\\\quad\quad\quad=\tan^2\alpha+1=VT.\\\Rightarrow 1+\tan^2\alpha=\dfrac{1}{\cos^2\alpha}\\b)1+\cot^2\alpha=\dfrac{1}{\sin^2\alpha}\\\text{Xét VP=}\dfrac{1}{\sin^2\alpha}\\\quad\quad\quad=\dfrac{\sin^2\alpha+\cos^2\alpha}{\sin^2\alpha}\\\quad\quad\quad=\dfrac{\sin^2\alpha}{\sin^2\alpha}+\dfrac{\cos^2\alpha}{\sin^2\alpha}\\\quad\quad\quad=1+\cot^2\alpha=VT.\\\Rightarrow 1+\cot^2\alpha=\dfrac{1}{\sin^2\alpha}\\\text{Những công thức đã dùng để chứng minh:}\\\sin^2\alpha+\cos^2\alpha=1;\dfrac{\sin\alpha}{\cos\alpha}=\tan\alpha;\dfrac{\cos\alpha}{\sin\alpha}=\cot\alpha.\)

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