lượng giác

J

jet_nguyen

Gợi ý:
Câu 1:
$$\cos^{13} x + \sin^{14} x = 1$$ Do $\sin x,\cos x \in [-1,1]$ nên ta có:
$$\cos^{13}x \le \cos^2x, \sin^{14}x \le \sin^2x$$ Vậy: $$\cos^{13} x + \sin^{14} x \le \cos^2x+\sin^2x= 1$$ Suy ra: $$\cos^{13} x + \sin^{14} x \le 1$$ Dấu "=" xảy ra khi: $\left\{\begin{array}{1} \sin^{14}x=\sin^2x \\ \cos^{13}x=\cos^2x \end{array}\right.$
Câu 2:
$$\dfrac{\sin^{10}x+\cos^{10}x}{4}=\dfrac{\sin^{6}x+\cos^{6}x}{\sin^22x+4\cos^22x}$$ Ta có:
$$\dfrac{\sin^6x+\cos^6x}{\sin^22x+4\cos^22x}= \dfrac{1-3\sin^2x.\cos^2x}{4-3\sin^22x}= \dfrac{1}{4}$$ Suy ra $VP=\dfrac{1}{4}$
Mà: $\cos^{10}x\le \cos^2x,\sin^{10}x\le \sin^2x$. Suy ra $VT \le \dfrac{\sin^2x+\cos^2x}{4} = \dfrac{1}{4}=VP$
Vậy: $VP=VT$ khi: $\left\{\begin{array}{1} \cos^{10}x=\cos^2x \\ \sin^{10}x=\sin^2x\end{array}\right.$

 
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