三角比を含む式の値

問題

　$\tan^235^\circ \sin^255^\circ+\tan^255^\circ \sin^235^\circ+(1+\tan^235^\circ)\sin^255^\circ$の値を求めよ．

【日本工業大学　2011】

解答

➀ $\tan^235^\circ \sin^255^\circ $，➁ $\tan^255^\circ \sin^235^\circ $，➂ $(1+\tan^235^\circ)\sin^255^\circ $に分けて計算する．

➀$\tan^235^\circ \sin^255^\circ =(\tan^235^\circ \sin^255^\circ)^2=\{\tan 35^\circ \sin(90^\circ -35^\circ)\}^2$

$=(\tan35^\circ \cos35^\circ)^2=(\dfrac{\sin 35^\circ}{\cos 35^\circ}\cdot \cos35^\circ)^2=\sin^2 35^\circ$

➁$\tan^255^\circ \sin^235^\circ =(\tan^255^\circ \sin^235^\circ)^2=\{\tan (90^\circ-35^\circ) \sin35^\circ\}^2$

$=(\dfrac{1}{\tan35^\circ} \sin35^\circ)^2=(\dfrac{\cos 35^\circ}{\sin 35^\circ}\cdot \sin35^\circ)^2=\cos^2 35^\circ$

➂$(1+\tan^235^\circ)\sin^255^\circ=\dfrac{1}{\cos^235^\circ}\cdot\sin^2(90^\circ-35^\circ)$

$=\dfrac{1}{\cos^235^\circ}\cdot \cos^235^\circ=1$

よって

　 $\tan^235^\circ \sin^255^\circ+\tan^255^\circ \sin^235^\circ+(1+\tan^235^\circ)\sin^255^\circ$

$=\sin^235^\circ+\cos^235^\circ+1$

$=1+1$

$=2$