tan a=2

=>sin a=2*cosa

\(P=\dfrac{10cosa-3cosa}{cosa+2\cdot2cosa}=\dfrac{7}{5}\)

Ta có: \(tan\alpha=2\Leftrightarrow\dfrac{sin\alpha}{cos\alpha}=2\Leftrightarrow sin\alpha=2cos\alpha\)

A = \(\dfrac{16cos^2\alpha+6cos^2\alpha}{20cos^2\alpha-2cos^2\alpha}=\dfrac{22cos^2\alpha}{18cos^2\alpha}=\dfrac{11}{9}\)

\(A=\dfrac{\dfrac{3sina}{cosa}-\dfrac{5cosa}{cosa}}{\dfrac{5sina}{cosa}+\dfrac{8cosa}{cosa}}=\dfrac{3tana-5}{5tana+8}=\dfrac{3.\left(\dfrac{5}{7}\right)-5}{5.\left(\dfrac{5}{7}\right)+8}=...\)

a, Ta có: 

\(\left(2^{\alpha}+2^{-\alpha}\right)^2\\ =\left(2^{\alpha}\right)^2+2\cdot2^{\alpha}\cdot2^{-\alpha}+\left(2^{-\alpha}\right)^2\\ =4^{\alpha}+4^{-\alpha}+2\\ =5+2\\ =7\)

Vậy \(2^{\alpha}+2^{-\alpha}=7\)

b, Ta có: 

\(4^{2\alpha}+4^{-2\alpha}=\left(4^{\alpha}\right)^2+\left(4^{-\alpha}\right)^2\\ =\left(4^{\alpha}+4^{-\alpha}\right)^2-2\\ =5^2-2\\ =23\)

Vậy \(4^{2\alpha}+4^{-2\alpha}=23\)

a)

$16^{\alpha }+16^{-\alpha } = (4^2)^{\alpha }+(4^2)^{-\alpha } = 4^{2\alpha }+4^{-2\alpha }$

$4^{2\alpha }+4^{-2\alpha } = 4^{2\log_4{\frac{1}{5}}}+4^{-2\log_4{\frac{1}{5}}} = \left(\frac{1}{5}\right)^2+\left(\frac{1}{5}\right)^{-2} = \frac{1}{25}+25 = \frac{26}{25}$

b)

$\left(2^{\alpha }+2^{-\alpha }\right)^2 = \left(\sqrt{4}\right)^{\alpha }+\left(\sqrt{4}\right)^{-\alpha } = 4^{\frac{\alpha}{2}}+4^{-\frac{\alpha}{2}}$

$4^{\frac{\alpha}{2}}+4^{-\frac{\alpha}{2}} = 4^{\frac{\log_4{\frac{1}{5}}}{2}}+4^{-\frac{\log_4{\frac{1}{5}}}{2}} = \left(\frac{1}{5}\right)^{\frac{1}{2}}+\left(\frac{1}{5}\right)^{-\frac{1}{2}} = \sqrt{\frac{1}{5}}+\frac{1}{\sqrt{5}} = \frac{2}{\sqrt{5}}$

\(\dfrac{2sina+cosa}{2sin^3a-cos^3a}=\dfrac{\dfrac{2sina}{cos^3a}+\dfrac{cosa}{cos^3a}}{\dfrac{2sin^3a}{cos^3a}-\dfrac{cos^3a}{cos^3a}}=\dfrac{2tana.\dfrac{1}{cos^2a}+\dfrac{1}{cos^2a}}{2tan^3a-1}\)

\(=\dfrac{2tana\left(1+tan^2a\right)+1+tan^2a}{2tan^3a-1}=...\) (thay số và bấm máy)

Lời giải:
Đặt $\frac{b}{2}=m; \frac{a}{2}=n$

Ta có:

$\tan m=4\tan n$.

$\tan (m-n)=\frac{\tan m-\tan n}{1+\tan m\tan n}=\frac{3\tan n}{1+4\tan ^2n}$

....

Thực ra nó chả ra một con số cụ thể nào cả, và cũng có nhiều kết quả biến đổi. Có lẽ bạn viết thiếu đề.

\(M=\frac{\sin^3a+3\cos^3a}{27\sin^3a-25\cos^3a}\)

\(M=\frac{\frac{\sin^3a+3\cos^3a}{\cos^3a}}{\frac{27\sin^3a-25\cos^3a}{\cos^3a}}\)

\(M=\frac{\tan^3a+3}{27\tan^3a-25}\)

\(M=\frac{\frac{8}{27}+3}{27.\frac{8}{27}-25}\)

\(M=\frac{\frac{89}{27}}{-17}\)

\(M=-\frac{89}{459}\)

P/s haphuong

Lời giải:
\(M=\frac{\frac{\sin a}{\cos a}+1}{\frac{\sin a}{\cos a}-1}=\frac{\tan a+1}{\tan a-1}=\frac{\frac{3}{5}+1}{\frac{3}{5}-1}=-4\)

\(N = \frac{\frac{\sin a\cos a}{\cos ^2a}}{\frac{\sin ^2a-\cos ^2a}{\cos ^2a}}=\frac{\frac{\sin a}{\cos a}}{(\frac{\sin a}{\cos a})^2-1}=\frac{\tan a}{\tan ^2a-1}=\frac{\frac{3}{5}}{\frac{3^2}{5^2}-1}=\frac{-15}{16}\)