35a) $A = (\sin\alpha +\cos\alpha)^2 + (\sin\alpha - \cos\alpha)^2$
$\to A = \sin^2\alpha + 2\sin\alpha\cos\alpha + \cos^2\alpha + \sin^2\alpha - 2\sin\alpha\cos\alpha + \cos^2\alpha$
$\to A = 2(\sin^2\alpha + \cos^2\alpha) = 2$
b) $B = \cos^6\alpha + \sin^6\alpha + 3\sin^2\alpha\cos^2\alpha$
$\to B = (\sin^2\alpha + \cos^2\alpha)^3 - 3\sin^2\alpha\cos^2\alpha(\sin^2\alpha + \cos^2\alpha) + 3\sin^2\alpha\cos^2\alpha$
$\to B = 1$
53a) $A= \sin^214^o + \sin^276^o + \tan1^o.\tan89^o - \dfrac{2\sin55^o}{\cos35^o}$
$\to A = \cos^276^o + \sin^276^o + \cot89^o.\tan89^o - \dfrac{2\cos35^o}{\cos35^o}$
$\to A = 1 + 1 - 2 = 0$
b) $B =\tan67^o + \cos^216^o - \cot23^o + \cos74^o -\dfrac{\cot37^o}{\tan53^o}$
$\to B = (\tan67^o - \cot23^o) + (\cos^216^o + \cos^74^o) - \dfrac{\cot37^o}{\tan53^o}$
$\to B = (\tan67^o - \tan67^o) + (\sin^274^o + \cos^74^o) - \dfrac{\tan53^o}{\tan53^o}$
$\to B = 0 + 1 - 1 = 0$
2.9) Xét $∆ABH$ và $∆ACK$ có:
$\widehat{A}:$ góc chung
$\widehat{H} = \widehat{K} = 90^o$
Do đó $∆ABH\sim ∆ACK\, (g.g)$
$\Rightarrow \dfrac{AB}{AC} = \dfrac{AH}{AK}$
$\Rightarrow \dfrac{AH}{AB} = \dfrac{AK}{AC}$
Xét $∆AHK$ và $∆ABC$ có:
$\widehat{A}:$ góc chung
$\dfrac{AH}{AB} = \dfrac{AK}{AC}$ $(cmt)$
Do đó $∆AHK\sim ∆ABC\, (c.g.c)$
$\Rightarrow \dfrac{S_{AHK}}{S_{ABC}} = \left(\dfrac{AH}{AB}\right)^2 = \sin^2\widehat{A} = \sin^230^o = \dfrac{1}{4}$
$\Rightarrow S_{ABC} = 4S_{AHK}$
$\Rightarrow S_{ABC} = 3S_{BKHC}$
