\(\left(cota+tana\right)^2-\left(cota-tana\right)^2\)

\(=cot^2a+tan^2a+2tana.cota-cot^2a-tan^2a+2tana.cota\)

\(=4tana.cota=4\)

Lời giải:

Sử dụng các công thức sau:

\(\bullet \tan \alpha=\frac{1}{\cot \alpha}\)

\(\bullet \tan (\alpha+\beta)=\frac{\tan \alpha+\tan \beta}{1-\tan\alpha.\tan \beta}\)

Ta có:

\(\text{VT}=\frac{1}{\tan a+\tan b}-\frac{1}{\cot a+\cot b}=\frac{1}{\tan a+\tan b}-\frac{1}{\frac{1}{\tan a}+\frac{1}{\tan b}}\)

\(=\frac{1}{\tan a+\tan b}-\frac{\tan a\tan b}{\tan a+\tan b}=\frac{1-\tan a\tan b}{\tan a+\tan b}\)

\(=\frac{1}{\frac{\tan a+\tan b}{1-\tan a\tan b}}=\frac{1}{\tan (a+b)}=\cot (a+b)=\text{VP}\)

Ta có đpcm.

\(sina=\frac{3}{5}\Rightarrow sin^2a=\frac{9}{25}\) ; \(cos^2a=1-\frac{9}{25}=\frac{16}{25}\)

\(A=\frac{cota+tana}{cota-tana}=\frac{sina.cosa\left(cota+tana\right)}{sina.cosa\left(cota-tana\right)}=\frac{cos^2a+sin^2a}{cos^2a-sin^2a}=\frac{1}{cos^2a-sin^2a}=\frac{1}{\frac{16}{25}-\frac{9}{25}}=\frac{25}{7}\)

\(B=\frac{sin^2a-cos^2a}{sin^2a-3cos^2a}=\frac{\frac{sin^2a}{sin^2a}-\frac{cos^2a}{sin^2a}}{\frac{sin^2a}{sin^2a}-\frac{3cos^2a}{sin^2a}}=\frac{1-cot^2a}{1-3cot^2a}=\frac{1-\left(-\frac{1}{3}\right)^2}{1-3\left(-\frac{1}{3}\right)^2}=\)

\(C_1=sin^2a+cos^2a+cos^2a=1+cos^2a=1+\frac{1}{1+tan^2a}=1+\frac{1}{1+\left(-2\right)^2}\)

\(C_2=\left(sin^2a+cos^2a\right)\left(sin^2a-cos^2a\right)=sin^2a-cos^2a=1-2cos^2a\)

\(=1-\frac{2}{1+tan^2a}=1-\frac{2}{1+\left(-2\right)^2}\)

Lời giải:

\((1+\cot a)\sin ^3a+(1+\tan a)\cos ^3a\)

\(=(1+\frac{\cos a}{\sin a})\sin ^3a+(1+\frac{\sin a}{\cos a})\cos ^3a\)

\(=(\sin a+\cos a)\sin ^2a+(\cos a+\sin a)\cos ^2a\)

\(=(\sin a+\cos a)(\sin ^2a+\cos ^2a)=(\sin a+\cos a).1=\sin a+\cos a\)

Ta có : \(\sin^2a+\cos^2a=1\Rightarrow\cos a=\frac{\sqrt{21}}{5}\)

Ta có : \(\frac{\cot a-\tan a}{\cot a+\tan a}=\frac{\frac{\cos a}{\sin a}-\frac{\sin a}{\cos a}}{\frac{\cos a}{\sin a}+\frac{\sin a}{\cos a}}\\ =\frac{\frac{\frac{\sqrt{21}}{5}}{\frac{2}{5}}-\frac{\frac{2}{5}}{\frac{\sqrt{21}}{5}}}{\frac{\frac{\sqrt{21}}{5}}{\frac{2}{5}}+\frac{\frac{2}{5}}{\frac{\sqrt{21}}{5}}}=\frac{17}{25}=0,68\)

Chọn C.

Ta có: cota + tana) ² = cot²a + 2.cota.tana + tan²a

= (cot²a + 1) + (tan²a + 1)