Question

Hãy đơn giản các biểu thức:

a) 1-sin²α

b) (1-cosα)(1+cosα)

c) 1+cos²α+sin²α

d) sinα-sinα cos²α

e) sin⁴α+cos⁴α+2sin²α cos²α

f) tan²α-sin²α tan²α

g) cos²α+tan²α cos²α

h) tan²α (2cos²α+sin²α-1)

Answer 1

a) 1- \(sin^2\alpha\)= \(cos^2\alpha\)

b) (\(1-cos\alpha\))(\(1+cos\alpha\)) = 1 - cos²\(\alpha\) = sin²\(\alpha\)

c) 1 + cos²\(\alpha\) + sin²\(\alpha\) = \(1+1=2\)

d) sin\(\alpha\) - sin\(\alpha.cos^2\alpha\)

= \(sin\alpha\left(1-cos^2\alpha\right)=sin\alpha.sin^2\alpha=sin^3\alpha\)

e) \(sin^4\alpha+cos^4\alpha+2sin^2\alpha.cos^2\alpha\)

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

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

f) \(tan^2\alpha-sin^2\alpha.tan^2\alpha\)

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

g) \(cos^2\alpha+tan^2\alpha.cos^2\alpha\)

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

h) \(tan^2\alpha\left(2cos^2\alpha+sin^2\alpha-1\right)\)

= \(tan^2\alpha\left[cos^2\alpha+\left(cos^2\alpha+sin^2\alpha\right)-1\right]\)

= \(tan^2\alpha\left(cos^2\alpha+1-1\right)\)

= \(tan^2\alpha.cos^2\alpha=sin^2\alpha\)