a.
\(cos85^0+cos35^0+cos25^0=2cos60^0.cos25^0+cos25^0\)
\(=cos25^0+cos25^0=2cos25^0\ne0\)
Đề bài sai, đề đúng: \(cos85^0+cos35^0-cos25^0=0\)
b.
\(tan9^0-tan27^0-tan63^0+tan81^0\)
\(=tan9^0+tan81^0-\left(tan27^0+tan63^0\right)\)
\(=tan9^0+cot9^0-\left(tan27^0+cot27^0\right)\)
\(=\dfrac{sin9^0}{cos9^0}+\dfrac{cos9^0}{sin9^0}-\left(\dfrac{sin27}{cos27}+\dfrac{cos27}{sin27}\right)\)
\(=\dfrac{sin^29+cos^29}{cos9.sin9}-\dfrac{sin^227+cos^227}{sin27.cos27}\)
\(=\dfrac{1}{\dfrac{1}{2}sin18}-\dfrac{1}{\dfrac{1}{2}sin54}=\dfrac{2}{sin18}-\dfrac{2}{sin54}\)
\(=\dfrac{2\left(sin54-sin18\right)}{sin54.sin18}=\dfrac{4cos36.sin18}{sin54.sin18}=\dfrac{4cos36}{sin54}=\dfrac{4cos36}{sin36}=4\)
c.
\(cos24+cos48-\left(cos84+cos12\right)\)
\(=2cos36.cos12-2cos36.cos48\)
\(=2cos36\left(cos12-cos48\right)\)
\(=4cos36.sin30.sin18\)
\(=\dfrac{2sin18.cos36.cos18}{cos18}=\dfrac{sin36.cos36}{cos18}\)
\(=\dfrac{sin72}{2cos18}=\dfrac{cos18}{2cos18}=\dfrac{1}{2}\)
d.
\(cosx-\dfrac{1}{2}cos3x-\dfrac{1}{2}cos5x\)
\(=cosx-\dfrac{1}{2}\left(cos3x+cos5x\right)\)
\(=cosx-cos4x.cosx\)
\(=cosx\left(1-cos4x\right)\)
\(=cosx\left(1-\left(1-2sin^22x\right)\right)\)
\(=2sin^22x.cosx\)
\(=2\left(2sinx.cosx\right)^2.cosx\)
\(=8sin^2x.cos^3x\)
