Упростите выражение:\newline $$1)\quad \sin^4 \alpha + 2\sin^2 \alpha \cos^2\alpha + \cos^4\alpha; \qquad 3) \quad \cos^4 \alpha - \cos^2 \alpha + \sin^2 \alpha;$$ $$2)\quad \frac{\sin \alpha}{1 + \cos \alpha} + \frac{1 + \cos \alpha}{\sin \alpha}; \qquad 4) \quad \sin^4 \alpha + \sin^2\cos^2\alpha + \cos^2\alpha.$$

\begin{enumerate} \item $$\sin^4 \alpha + 2\sin^2 \alpha \cos^2\alpha + \cos^4\alpha = (\sin^2\alpha + \cos^2\alpha)^2 = 1;$$ \item $$\frac{\sin \alpha}{1 + \cos \alpha} + \frac{1 + \cos \alpha}{\sin \alpha} = \frac{\sin^2\alpha + 1 + 2\cos \alpha + \cos^2\alpha}{\sin\alpha(1 + \cos \alpha)} = \frac{2 + 2\cos \alpha}{\sin\alpha(1 + \cos \alpha)} = \frac{2(1 + \cos \alpha)}{\sin \alpha(1 + \cos \alpha)} = \frac{2}{\sin \alpha}$$ \item $$\cos^4 \alpha - \cos^2 \alpha + \sin^2 \alpha = \cos^2(\cos^2\alpha - 1) + \sin^2\alpha = \cos^2\alpha\cdot(-\sin^2\alpha) + \sin^2 \alpha = \sin^2\alpha(-\cos^2\alpha + 1) = \sin^4\alpha.$$ \item $$\sin^4 \alpha + \sin^2\cos^2\alpha + \cos^2\alpha = \sin^2\alpha(\sin^2\alpha + \cos^2\alpha) + \cos^2\alpha = \sin^2 \alpha + \cos^2\alpha = 1.$$ \end{enumerate}

Homepage › Solution manuals › Аркадий Мерзляк › Алгебра и начала анализа, геометрия. 10 класс › Задание 12.6

Алгебра и начала анализа, геометрия. 10 класс

Задание 12.6

Упростите выражение:

1) \sin^{4} α + 2 \sin^{2} α \cos^{2} α + \cos^{4} α; 3) \cos^{4} α - \cos^{2} α + \sin^{2} α;

2) \frac{\sin α}{1 + \cos α} + \frac{1 + \cos α}{\sin α}; 4) \sin^{4} α + \sin^{2} \cos^{2} α + \cos^{2} α .

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gimli

2021-11-13 06:08

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1.: $\sin^{4} α + 2 \sin^{2} α \cos^{2} α + \cos^{4} α = {(\sin^{2} α + \cos^{2} α)}^{2} = 1;$
2.: $\frac{\sin α}{1 + \cos α} + \frac{1 + \cos α}{\sin α} = \frac{\sin^{2} α + 1 + 2 \cos α + \cos^{2} α}{\sin α (1 + \cos α)} = \frac{2 + 2 \cos α}{\sin α (1 + \cos α)} = \frac{2 (1 + \cos α)}{\sin α (1 + \cos α)} = \frac{2}{\sin α}$
3.: $\cos^{4} α - \cos^{2} α + \sin^{2} α = \cos^{2} (\cos^{2} α - 1) + \sin^{2} α = \cos^{2} α \cdot (- \sin^{2} α) + \sin^{2} α = \sin^{2} α (- \cos^{2} α + 1) = \sin^{4} α .$
4.: $\sin^{4} α + \sin^{2} \cos^{2} α + \cos^{2} α = \sin^{2} α (\sin^{2} α + \cos^{2} α) + \cos^{2} α = \sin^{2} α + \cos^{2} α = 1 .$

gimli

2021-11-13 06:08

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gimli

2021-11-13 06:08

Задание 12.6

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