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Câu hỏi: Không sử dụng bảng số và máy tính, hãy tính
a) $\sin ^{4} \frac{\pi}{16}+\sin ^{4} \frac{3 \pi}{16}+\sin ^{4} \frac{5 \pi}{16}+\sin ^{4} \frac{7 \pi}{16}$
b) $\cot 7,5^{\circ}+\tan 67,5^{\circ}-\tan 7,5^{\circ}-\cot 67,5^{\circ}$
a) $\sin ^{4} \frac{\pi}{16}+\sin ^{4} \frac{3 \pi}{16}+\sin ^{4} \frac{5 \pi}{16}+\sin ^{4} \frac{7 \pi}{16}$
b) $\cot 7,5^{\circ}+\tan 67,5^{\circ}-\tan 7,5^{\circ}-\cot 67,5^{\circ}$
a)
$\sin ^{4} \frac{\pi}{16}+\sin ^{4} \frac{3 \pi}{16}+\sin ^{4} \frac{5 \pi}{16}+\sin ^{4} \frac{7 \pi}{16}$
$=\left(\frac{1-\cos \frac{\pi}{8}}{2}\right)^{2}+\left(\frac{1-\cos \frac{3 \pi}{8}}{2}\right)^{2}+\left(\frac{1-\cos \frac{5 \pi}{8}}{2}\right)^{2}+\left(\frac{1-\cos \frac{7 \pi}{8}}{2}\right)^{2}$
$
\begin{aligned}=\frac{1}{4}\left(1-2 \cos \frac{\pi}{8}+\cos ^{2} \frac{\pi}{8}+1-2 \cos \frac{3 \pi}{8}+\cos ^{2} \frac{3 \pi}{8}+1-2 \cos \frac{5 \pi}{8}+\cos ^{2} \frac{5 \pi}{8}\right.\\\left.+1-2 \cos \frac{7 \pi}{\dot{8}}+\cos ^{2} \frac{7 \pi}{8}\right) \end{aligned}
$
$=1-\frac{1}{2}\left(\cos \frac{\pi}{8}+\cos \frac{3 \pi}{8}+\cos \frac{5 \pi}{8}+\cos \frac{7 \pi}{8}\right)$
$+\frac{1}{4}\left(\frac{1+\cos \frac{\pi}{4}}{2}+\frac{1+\cos \frac{3 \pi}{4}}{2}+\frac{1+\cos \frac{5 \pi}{4}}{2}+\frac{1+\cos \frac{7 \pi}{4}}{2}\right)$
$=1-\frac{1}{2}\left(\cos \frac{\pi}{8}+\cos \frac{3 \pi}{8}-\cos \frac{3 \pi}{8}-\cos \frac{\pi}{8}\right)+\frac{1}{8}\left(4+\frac{\sqrt{2}}{2}-\frac{\sqrt{2}}{2}-\frac{\sqrt{2}}{2}+\frac{\sqrt{2}}{2}\right)$
$=\frac{3}{2} .$
b)
$\cot 7,5^{\circ}+\tan 67,5^{\circ}-\tan 7,5^{\circ}-\cot 67,5^{\circ}$
$=\frac{\cos 7,5^{\circ}}{\sin 7,5^{\circ}}-\frac{\sin 7,5^{\circ}}{\cos 7,5^{\circ}}+\frac{\sin 67,5^{\circ}}{\cos 67,5^{\circ}}-\frac{\cos 67,5^{\circ}}{\sin 67,5^{\circ}}$
$=\frac{\cos ^{2} 7,5^{\circ}-\sin ^{2} 7,5^{\circ}}{\sin 7,5^{\circ} \cos 7,5^{\circ}}+\frac{\sin ^{2} 67.5^{\circ}-\cos ^{2} 67,5^{\circ}}{\sin 67,5^{\circ} \cos 67,5^{\circ}}$
$=\frac{\cos 15^{\circ}}{\frac{1}{2} \sin 15^{\circ}}-\frac{\cos 135^{\circ}}{\frac{1}{2} \sin 135^{\circ}}=\frac{2\left(\sin 135^{\circ} \cos 15^{\circ}-\cos 135^{\circ} \sin 15^{\circ}\right)}{\sin 15^{\circ} \sin 135^{\circ}}$
$=\frac{2 \sin \left(135^{\circ}-15^{\circ}\right)}{\sin \left(45^{\circ}-30^{\circ}\right) \sin \left(180^{\circ}-45^{\circ}\right)}$
$=\frac{2 \sin 120^{\circ}}{\left(\sin 45^{\circ} \cos 30^{\circ}-\cos 45^{\circ} \sin 30^{\circ}\right) \sin 45^{\circ}}$
$=\frac{\sqrt{3}}{\frac{\sqrt{2}}{2}\left(\frac{\sqrt{3}}{2}-\frac{1}{2}\right) \cdot \frac{\sqrt{2}}{2}}=\frac{4 \sqrt{3}}{\sqrt{3}-1}=6+2 \sqrt{3} .$
$\sin ^{4} \frac{\pi}{16}+\sin ^{4} \frac{3 \pi}{16}+\sin ^{4} \frac{5 \pi}{16}+\sin ^{4} \frac{7 \pi}{16}$
$=\left(\frac{1-\cos \frac{\pi}{8}}{2}\right)^{2}+\left(\frac{1-\cos \frac{3 \pi}{8}}{2}\right)^{2}+\left(\frac{1-\cos \frac{5 \pi}{8}}{2}\right)^{2}+\left(\frac{1-\cos \frac{7 \pi}{8}}{2}\right)^{2}$
$
\begin{aligned}=\frac{1}{4}\left(1-2 \cos \frac{\pi}{8}+\cos ^{2} \frac{\pi}{8}+1-2 \cos \frac{3 \pi}{8}+\cos ^{2} \frac{3 \pi}{8}+1-2 \cos \frac{5 \pi}{8}+\cos ^{2} \frac{5 \pi}{8}\right.\\\left.+1-2 \cos \frac{7 \pi}{\dot{8}}+\cos ^{2} \frac{7 \pi}{8}\right) \end{aligned}
$
$=1-\frac{1}{2}\left(\cos \frac{\pi}{8}+\cos \frac{3 \pi}{8}+\cos \frac{5 \pi}{8}+\cos \frac{7 \pi}{8}\right)$
$+\frac{1}{4}\left(\frac{1+\cos \frac{\pi}{4}}{2}+\frac{1+\cos \frac{3 \pi}{4}}{2}+\frac{1+\cos \frac{5 \pi}{4}}{2}+\frac{1+\cos \frac{7 \pi}{4}}{2}\right)$
$=1-\frac{1}{2}\left(\cos \frac{\pi}{8}+\cos \frac{3 \pi}{8}-\cos \frac{3 \pi}{8}-\cos \frac{\pi}{8}\right)+\frac{1}{8}\left(4+\frac{\sqrt{2}}{2}-\frac{\sqrt{2}}{2}-\frac{\sqrt{2}}{2}+\frac{\sqrt{2}}{2}\right)$
$=\frac{3}{2} .$
b)
$\cot 7,5^{\circ}+\tan 67,5^{\circ}-\tan 7,5^{\circ}-\cot 67,5^{\circ}$
$=\frac{\cos 7,5^{\circ}}{\sin 7,5^{\circ}}-\frac{\sin 7,5^{\circ}}{\cos 7,5^{\circ}}+\frac{\sin 67,5^{\circ}}{\cos 67,5^{\circ}}-\frac{\cos 67,5^{\circ}}{\sin 67,5^{\circ}}$
$=\frac{\cos ^{2} 7,5^{\circ}-\sin ^{2} 7,5^{\circ}}{\sin 7,5^{\circ} \cos 7,5^{\circ}}+\frac{\sin ^{2} 67.5^{\circ}-\cos ^{2} 67,5^{\circ}}{\sin 67,5^{\circ} \cos 67,5^{\circ}}$
$=\frac{\cos 15^{\circ}}{\frac{1}{2} \sin 15^{\circ}}-\frac{\cos 135^{\circ}}{\frac{1}{2} \sin 135^{\circ}}=\frac{2\left(\sin 135^{\circ} \cos 15^{\circ}-\cos 135^{\circ} \sin 15^{\circ}\right)}{\sin 15^{\circ} \sin 135^{\circ}}$
$=\frac{2 \sin \left(135^{\circ}-15^{\circ}\right)}{\sin \left(45^{\circ}-30^{\circ}\right) \sin \left(180^{\circ}-45^{\circ}\right)}$
$=\frac{2 \sin 120^{\circ}}{\left(\sin 45^{\circ} \cos 30^{\circ}-\cos 45^{\circ} \sin 30^{\circ}\right) \sin 45^{\circ}}$
$=\frac{\sqrt{3}}{\frac{\sqrt{2}}{2}\left(\frac{\sqrt{3}}{2}-\frac{1}{2}\right) \cdot \frac{\sqrt{2}}{2}}=\frac{4 \sqrt{3}}{\sqrt{3}-1}=6+2 \sqrt{3} .$