Question

CMR:

\(a.tan^2\alpha+1=\dfrac{1}{cos^2\alpha}\)

b)\(cot^2\alpha+1=\dfrac{1}{sin^2\alpha}\)

c)\(cos^4\alpha-sin^4\alpha=2cos^2\alpha-1\)

Answer 1

a) \(tan^2\alpha+1=\dfrac{1}{cos^2\alpha}\)

Xét vế trái ta có:

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

Xét vế trái ta có:

\(VT=cot^2\alpha+1=\dfrac{cos^2\alpha}{sin^2\alpha}+1=\dfrac{cos^2\alpha+sin^2\alpha}{sin^2\alpha}=\dfrac{1}{sin^2\alpha}=VP\left(đpcm\right)\)

c) \(cos^4\alpha-sin^4\alpha=2cos^2\alpha-1\)

Xét vế trái ta có:

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