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\(sin^2\text{α}=1-cos^2\text{α}=1-\left(\dfrac{1}{3}\right)^2=\dfrac{8}{9}\)
vì π<α<\(\dfrac{\Pi}{2}\)⇒sin α=\(\dfrac{2\sqrt{2}}{3}\)
do a ∈ \(\left(0;\dfrac{\pi}{2}\right)\)⇒ \(\left\{{}\begin{matrix}sinx>0\\cosx>0\end{matrix}\right.\)
Mà tanx = 3 ⇒ \(\dfrac{sinx}{cosx}=3\Leftrightarrow\dfrac{sin^2x}{cos^2x}=9\Rightarrow10sin^2x=9\)
⇒ sinx = \(\dfrac{3}{\sqrt{10}}\)
⇒ sin (x + π) = -sinx = -\(\dfrac{3}{\sqrt{10}}\)
\(P=cos2a=1-2sin^2a=1-2.\left(\dfrac{4}{5}\right)^2=-\dfrac{7}{25}\)
a, \(\dfrac{1-sin2a}{1+sin2a}\)
\(=\dfrac{sin^2a+cos^2a-2sina.cosa}{sin^2a+cos^2a+2sina.cosa}\)
\(=\dfrac{\left(sina-cosa\right)^2}{\left(sina+cosa\right)^2}\)
\(=\dfrac{2sin^2\left(a-\dfrac{\pi}{4}\right)}{2sin^2\left(a+\dfrac{\pi}{4}\right)}\)
\(=\dfrac{sin^2\left(\dfrac{\pi}{4}-a\right)}{sin^2\left(a+\dfrac{\pi}{4}\right)}\)
\(=\dfrac{cos^2\left(\dfrac{\pi}{4}+a\right)}{sin^2\left(\dfrac{\pi}{4}+a\right)}=cot\left(\dfrac{\pi}{4}+a\right)\)
b, \(\dfrac{sina+sinb.cos\left(a+b\right)}{cosa-sinb.sin\left(a+b\right)}\)
\(=\dfrac{sina+sinb.cosa.cosb-sinb.sina.sinb}{cosa-sinb.sina.cosb-sinb.cosa.sinb}\)
\(=\dfrac{sina.\left(1-sin^2b\right)+sinb.cosa.cosb}{cosa.\left(1-sin^2b\right)-sinb.sina.cosb}\)
\(=\dfrac{sina.cos^2b+sinb.cosa.cosb}{cosa.cos^2b-sinb.sina.cosb}\)
\(=\dfrac{\left(sina.cosb+sinb.cosa\right).cosb}{\left(cosa.cosb-sinb.sina\right).cosb}\)
\(=\dfrac{sin\left(a+b\right)}{cos\left(a+b\right)}=tan\left(a+b\right)\)
\(A=sin\left(\dfrac{5\pi}{2}-\alpha\right)-cos\left(\dfrac{13\pi}{2}-\alpha\right)-3sin\left(\alpha-5\pi\right)-2sin\alpha-cos\alpha\)
\(=sin\left(\dfrac{\pi}{2}-\alpha\right)-cos\left(\dfrac{\pi}{2}-\alpha\right)-3sin\left(\alpha-\pi\right)-2sin\alpha-cos\alpha\)
\(=cos\alpha-sin\alpha+3sin\left(\pi-\alpha\right)-2sin\alpha-cos\alpha\)
\(=cos\alpha-sin\alpha+3sin\alpha-2sin\alpha-cos\alpha=0\)
\(\dfrac{3\pi}{2}\le a\le2\pi\Rightarrow3\pi\le2a\le4\pi\)
\(\Rightarrow sin2a\le0\)
\(cos^2a-sin^2a=\dfrac{1}{2}\Leftrightarrow cos2a=\dfrac{1}{2}\)
\(\Rightarrow sin2a=-\sqrt{1-cos^22a}=-\dfrac{\sqrt{3}}{2}\)
Vì 0 < α < π/2 nên sin α > 0, cos α > 0, tan α > 0, cot α > 0.
`\pi/2 < \alpha < \pi=>\alpha` nằm ở góc phần tư thứ `2`
`=>{(sin \alpha > 0;cos \alpha < 0),(tan \alpha < 0; cot \alpha < 0):}`
`->\bb D`