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ta có : \(M=2cot37.cot53+sin^228\dfrac{3tan54}{cot36}+sin^262\)
\(=2.cot37.cot\left(90-37\right)+sin^228\dfrac{3tan54}{cot\left(90-54\right)}+sin^262\)
\(=2.cot37.tan37+sin^228\dfrac{3tan54}{tan54}+sin^262\)\(=2+3sin^228+sin^262=2+2sin^228+sin^228+sin^2\left(90-28\right)\)
\(=2+2sin^228+sin^228+cos^228=3+2sin^228\)
a) 1 + tan22 a =1 +(\(\dfrac{sina}{cosa}\))2 =\(\dfrac{sina+cosa}{cos^2a}\)=\(\dfrac{1}{cos^2a}\)
b) 1 + cot2 a= 1 +(\(\dfrac{cosa}{sina}\))2 = \(\dfrac{cosa+sina}{sin^2a}\)=\(\dfrac{1}{sin^2a}\)
c) tan2 a (2 sin2a + 3 cos2 a - 2)
=tan2 a[cos2 a +2 (\(sina^2+cos^2a\))-2 ]
=\(\dfrac{sin^2a}{cos^2a}\)×\(cos^2a=sin^2a\)
b: \(1+cot^2a=1+\left(\dfrac{cosa}{sina}\right)^2=\dfrac{1}{sin^2a}\)
c: \(=tan^2a\left[2\left(1-cos^2a\right)+3cos^2a-2\right]\)
\(=tan^2a\left[cos^2a\right]\)
\(=\dfrac{sin^2a}{cos^2a}\cdot cos^2a=sin^2a\)
Ta có : \(\cot\left(37\right)=\tan\left(53\right)\) ,\(\sin^2\alpha+\cos^2\alpha=1,\tan\alpha\cdot\cot\alpha=1\)
\(sin\left(28\right)=\cos\left(62\right)\)
\(\Leftrightarrow sin^2\left(28\right)=\cos^2\left(62\right)\)
\(\cot\left(36\right)=\tan\left(54\right)\)
Đề : \(\cot\left(37\right)\cdot\cot\left(53\right)+\sin^2\left(28\right)-\frac{3\cdot\tan\left(54\right)}{\cot\left(36\right)}+sin^2\left(62\right)\)
\(=\tan\left(53\right)\cdot\cot\left(53\right)+\cos^2\left(62\right)-\frac{3\cdot\tan\left(54\right)}{\tan\left(54\right)}+\sin^2\left(62\right)\)
\(=\)\(\tan\left(53\right)\cdot\cot\left(53\right)+\cos^2\left(62\right)+\sin^2\left(62\right)-\frac{3\cdot\tan\left(54\right)}{\tan\left(54\right)}\)
\(=1+1-3\)
\(=-1\)
\(=2\cdot sin53^0\cdot cos53^0+1-3=sin106^0-2\)