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Hai góc phụ nhau thì sin góc nọ bằng cos góc kia, từ đó ta có: \(sin1^o=cos89^o,sin2^o=cos88^o,...\)
Từ đó ta suy ra \(A=cos^289^o+sin^289^o+cos^288^o+sin^288^o+...+sin^245^o\)
\(=1+1+...+\frac{1}{2}=44\frac{1}{2}\)
\(M=\left(\sin^210^0+\sin^280^0\right)+\left(\sin^220^0+\sin^270^0\right)-3\tan39^0\cdot\cot39^0\\ M=\left(\sin^210^0+\cos^210^0\right)+\left(\sin^220^0+\cos^220^0\right)-3\cdot1=1+1-3=-1\)
Lời giải:
\(M=\frac{\frac{\sin a}{\cos a}+1}{\frac{\sin a}{\cos a}-1}=\frac{\tan a+1}{\tan a-1}=\frac{\frac{3}{5}+1}{\frac{3}{5}-1}=-4\)
\(N = \frac{\frac{\sin a\cos a}{\cos ^2a}}{\frac{\sin ^2a-\cos ^2a}{\cos ^2a}}=\frac{\frac{\sin a}{\cos a}}{(\frac{\sin a}{\cos a})^2-1}=\frac{\tan a}{\tan ^2a-1}=\frac{\frac{3}{5}}{\frac{3^2}{5^2}-1}=\frac{-15}{16}\)
\(\left(\sqrt{\dfrac{1+sin\alpha}{1-sin\alpha}}+\sqrt{\dfrac{1-sin\alpha}{1+sin\alpha}}\right).\dfrac{1}{\sqrt{1+tan^2\alpha}}\)
\(=\left(\sqrt{\dfrac{\left(1+sin\alpha\right)^2}{\left(1-sin\alpha\right)\left(1+sin\alpha\right)}}+\sqrt{\dfrac{\left(1-sin\alpha\right)^2}{\left(1+sin\alpha\right)\left(1-sin\alpha\right)}}\right).\dfrac{1}{\sqrt{1+\left(\dfrac{sin\alpha}{cos\alpha}\right)^2}}\)
\(=\left(\sqrt{\dfrac{\left(1+sin\alpha\right)^2}{1-sin^2\alpha}}+\sqrt{\dfrac{\left(1-sin\alpha\right)^2}{1-sin^2\alpha}}\right).\dfrac{1}{\sqrt{\dfrac{cos^2\alpha+sin^2\alpha}{cos^2\alpha}}}\)
\(=\left(\sqrt{\dfrac{\left(1+sin\alpha\right)^2}{cos^2\alpha}}+\sqrt{\dfrac{\left(1-sin\alpha\right)^2}{cos^2\alpha}}\right).\dfrac{1}{\sqrt{\dfrac{1}{cos^2\alpha}}}\)
\(=\left(\dfrac{1+sin\alpha}{cos\alpha}+\dfrac{1-sin\alpha}{cos\alpha}\right).\dfrac{1}{\dfrac{1}{cos\alpha}}=\dfrac{2}{cos\alpha}.cos\alpha=2\)
\(1+tan^2a=\frac{1}{cos^2a}\)
\(1+3^2=\frac{1}{cos^2a}\)
\(10=\frac{1}{cos^2a}\)
\(cos^2a=\frac{1}{10}\)
\(cosa=\pm\sqrt{\frac{1}{10}}\)
\(sin^2a+cos^2a=1\)
\(sin^2a+\frac{1}{10}=1\)
\(sin^2a=\frac{9}{10}\)
\(sina=+\sqrt{\frac{9}{10}}\)
Vì tan dương nên có hai trường hợp :
TH1 : cả sin và cos cùng dương :
\(A=\frac{sina\cdot cosa}{sin^2a-cos^2a}\)
\(=\frac{\sqrt{\frac{9}{10}}\cdot\sqrt{\frac{1}{10}}}{\frac{9}{10}-\frac{1}{10}}\)
\(=\frac{\frac{3}{10}}{\frac{8}{10}}\)
\(=\frac{3}{8}\)
TH2 : cả sin và cos cùng âm
\(A=\frac{sina\cdot cosa}{sin^2a-cos^2a}\)
\(=\frac{-\sqrt{\frac{9}{10}}\cdot-\sqrt{\frac{1}{10}}}{\frac{9}{10}-\frac{1}{10}}\)
\(=\frac{\frac{3}{10}}{\frac{8}{10}}\)
\(=\frac{3}{8}\)
C = \(\frac{cosa-sina}{cosa+sina}=\frac{1-tana}{1+tana}=\frac{1-2}{1+2}=-\frac{1}{3}\)