Question

Câu 3: Chứng minh các đẳng thức lượng giác sau:

a) sin(60 độ+a) - sin(60 độ - a) = sin a

b) \(sin^4a+cos^4a\) = 3/4 + 1/4.cos4a

c) sina(2cos4a+2cos2a+1) = sin5a

d) \(\dfrac{cos\left(a-\beta\right)}{cos\left(a+\beta\right)}=\dfrac{1+tana.tan\beta}{1-tana.tan\beta}\)

Answer 1

a) \(sin\left(60^o+a\right)-sin\left(60^o-a\right)=sina\)

\(\Leftrightarrow2cos\dfrac{60^o+a+60^o-a}{2}.sin\dfrac{60^o+a-60^o+a}{2}=sina\)

\(\Leftrightarrow2cos60^o.sina=sina\)

\(\Leftrightarrow2.\dfrac{1}{2}sina=sina\left(đúng\right)\)

\(\Rightarrow dpcm\)

Answer 2

d) \(VT=\dfrac{cos\left(\alpha-\beta\right)}{cos\left(\alpha+\beta\right)}=\dfrac{cos\alpha cos\beta+sin\alpha sin\beta}{cos\alpha cos\beta-sin\alpha sin\beta}\)

\(\Leftrightarrow VT=\dfrac{cos\alpha cos\beta\left(1+\dfrac{sin\alpha sin\beta}{cos\alpha cos\beta}\right)}{cos\alpha cos\beta\left(1-\dfrac{sin\alpha sin\beta}{cos\alpha cos\beta}\right)}\)

\(\Leftrightarrow VT=\dfrac{1+tan\alpha.tan\beta}{1-tan\alpha.tan\beta}=VP\)

\(\Rightarrow dpcm\)

Answer 3

b) \(VT=sin^4a+cos^4a\)

\(\Leftrightarrow VT=\left(sin^2a+cos^2a\right)^2-2sin^2acos^2a\)

\(\Leftrightarrow VT=1-\dfrac{1}{2}\left(2sinacosa\right)^2\)

\(\Leftrightarrow VT=1-\dfrac{1}{2}sin^22a\)

\(\Leftrightarrow VT=\dfrac{3}{4}+\dfrac{1}{4}-\dfrac{1}{2}sin^22a\)

\(\Leftrightarrow VT=\dfrac{3}{4}+\dfrac{1}{4}\left(1-2sin^22a\right)\)

\(\Leftrightarrow VT=\dfrac{3}{4}+\dfrac{1}{4}.cos4a=VP\)

\(\Rightarrow dpcm\)