Question

Chứng minh với α là góc nhọn tùy ý

a, Cos⁴ α - Sin⁴ α = 1 - 2Sin² α

b, 1 + tan² α = 1/Cos² α

Answer 1

a) ta có \(VT=cos^4\alpha-sin^4\alpha=\left(cos^4\alpha+sin^4\alpha\right)-2sin^4\alpha\)

\(=\left(\left(sin^2\alpha+cos^2\alpha\right)^2-2sin^2\alpha.cos^2\alpha\right)-2sin^4\alpha\)

\(=1-2sin^2\alpha.cos^2\alpha-2sin^4\alpha=1-2sin^2\alpha\left(cos^2\alpha+sin^2\alpha\right)\)

\(=1-2sin^2\alpha=VP\left(đpcm\right)\)

b) ta có : \(VP=\dfrac{1}{cos^2\alpha}=\dfrac{sin^2\alpha+cos^2\alpha}{cos^2\alpha}=1+tan^2\alpha=VT\left(đpcm\right)\)