\(\dfrac{\left(cosa-sina\right)^2-\left(cosa+sina\right)^2}{cosa\cdot sina}\)

\(=\dfrac{\left(cosa-sina-cosa-sina\right)\left(cosa-sina+cosa+sina\right)}{cosa\cdot sina}\)

\(=\dfrac{-2\cdot sina\cdot2\cdot cosa}{cosa\cdot sina}=-4\)

\(=\frac{\left(\sin a+\cos a-\sin a+\cos a\right)\left(\sin a+\cos a+\sin a-\cos a\right)}{\sin a.\cos a}=\frac{2.\cos a.2.\sin a}{\sin a.\cos a}=4\)

ĂN CHO CÒN NÓNG:NGON.

\(sina+cosa=\frac{5}{4}\Rightarrow\left(sina+cosa\right)^2=\frac{25}{16}\)

\(\Rightarrow sin^2a+cos^2a+2sina.cosa=\frac{25}{16}\)

\(sina.cosa=\frac{\frac{25}{16}-1}{2}=\frac{9}{32}\)

b/ \(\left(sina-cosa\right)^2=sin^2a+cos^2a-2sinacosa\)

\(\left(sina-cosa\right)^2=1-2.\frac{9}{32}=\frac{7}{16}\)

\(\Rightarrow sina-cosa=\pm\frac{\sqrt{7}}{4}\)

c/ \(sin^3a-cos^3a=\left(sina-cosa\right)\left(sin^2a+cos^2a+sina.cosa\right)\)

\(=\left(sina-cosa\right)\left(1+\frac{9}{32}\right)=\pm\frac{41\sqrt{7}}{128}\)

a) √2 cos(x - π/4)

= √2.(cosx.cos π/4 + sinx.sin π/4)

= √2.(√2/2.cosx + √2/2.sinx)

= √2.√2/2.cosx + √2.√2/2.sinx

= cosx + sinx (đpcm)

b) √2.sin(x - π/4)

= √2.(sinx.cos π/4 - sin π/4.cosx )

= √2.(√2/2.sinx - √2/2.cosx )

= √2.√2/2.sinx - √2.√2/2.cosx

= sinx – cosx (đpcm).

b) khai triển hằng đẳng thức là ra

a) nhân tích chéo

\(\frac{\cos\alpha}{1-\sin\alpha}=\frac{1+\sin\alpha}{\cos\alpha}\Leftrightarrow\cos^2\alpha=1-\sin^2\alpha\)\(\Leftrightarrow\cos^2\alpha+\sin^2\alpha=1\)(luôn đúng)

\(\frac{\left(\sin\alpha+\cos\alpha\right)^2-\left(\sin\alpha-\cos\alpha\right)^2}{\sin\alpha\cdot\cos\alpha}=\frac{\sin^2\alpha+\cos^2\alpha+2\sin\alpha\cdot\cos\alpha-\sin^2\alpha-\cos^2\alpha+2\sin\alpha\cdot\cos\alpha}{\sin\alpha\cdot\cos\alpha}\)

\(=\frac{4\sin\alpha\cdot\cos\alpha}{\sin\alpha\cdot\cos\alpha}=4\)(đpcm)

a: \(\sin^2a+\cos^2a=1\)

\(\Leftrightarrow\cos^2a=1-\sin^2a=\left(1-\sin a\right)\left(1+\sin a\right)\)

hay \(\dfrac{\cos a}{1-\sin a}=\dfrac{1+\sin a}{\cos a}\)

b: \(VT=\dfrac{\left(\sin a+\cos a+\sin a-\cos a\right)\left(\sin a+\cos a-\sin a+\cos a\right)}{\sin a\cdot\cos a}\)

\(=\dfrac{2\cdot\cos a\cdot2\sin a}{\sin a\cdot\cos a}=4\)

\(sina+cosa=\dfrac{1}{2}\Rightarrow\left(sina+cosa\right)^2=\dfrac{1}{4}\Rightarrow2sinacosa=\dfrac{1}{4}-1=\dfrac{-3}{4}\)

\(\Leftrightarrow-2sinacosa=\dfrac{3}{4}\)

\(\Leftrightarrow cos^2a+sin^2a-2sinacosa=cos^2a+sin^2a+\dfrac{3}{4}\)

\(\Rightarrow\left(sina-cosa\right)^2=1+\dfrac{3}{4}=\dfrac{7}{4}\)

\(\Rightarrow\left|sina-cosa\right|=\dfrac{\sqrt{7}}{2}\)

cos(\(\dfrac{3a}{2}\))*cos(\(\dfrac{a}{2}\))=\(\dfrac{1}{2}\left(cos\left(\dfrac{3a}{2}+\dfrac{a}{2}\right)+cos\left(\dfrac{3a}{2}-\dfrac{a}{2}\right)\right)\)=\(\dfrac{1}{2}\left(cos\left(2a\right)+cos\left(a\right)\right)\)=\(\dfrac{1}{2}\left(2cos^2a-1+cosa\right)\)=\(\dfrac{1}{2}\left(2\cdot\left(\dfrac{3}{4}\right)^2-1+\dfrac{3}{4}\right)=\dfrac{7}{16}\)

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