与えられた3つの数式をそれぞれ簡単にします。 (1) $\{\sin(90^\circ - A) \tan A - \cos(90^\circ - A) \tan(90^\circ - A)\}^2 + 2 \sin A \sin(90^\circ - A)$ (2) $(\sin 40^\circ + \cos 40^\circ)^2 + (\sin 50^\circ - \cos 50^\circ)^2$ (3) $\sin 35^\circ \cos 35^\circ (\tan 35^\circ + \tan 55^\circ)$

1. 問題の内容

与えられた3つの数式をそれぞれ簡単にします。

(1) $\{\sin(90^\circ - A) \tan A - \cos(90^\circ - A) \tan(90^\circ - A)\}^2 + 2 \sin A \sin(90^\circ - A)$

(2) $(\sin 40^\circ + \cos 40^\circ)^2 + (\sin 50^\circ - \cos 50^\circ)^2$

(3) $\sin 35^\circ \cos 35^\circ (\tan 35^\circ + \tan 55^\circ)$

2. 解き方の手順

(1)

$\sin(90^\circ - A) = \cos A$

$\cos(90^\circ - A) = \sin A$

$\tan(90^\circ - A) = \frac{1}{\tan A}$

与式は

$\{\cos A \tan A - \sin A \frac{1}{\tan A}\}^2 + 2 \sin A \cos A$

$= \{\cos A \frac{\sin A}{\cos A} - \sin A \frac{\cos A}{\sin A}\}^2 + 2 \sin A \cos A$

$= (\sin A - \cos A)^2 + 2 \sin A \cos A$

$= \sin^2 A - 2 \sin A \cos A + \cos^2 A + 2 \sin A \cos A$

$= \sin^2 A + \cos^2 A$

$= 1$

(2)

$(\sin 40^\circ + \cos 40^\circ)^2 + (\sin 50^\circ - \cos 50^\circ)^2$

$= (\sin^2 40^\circ + 2 \sin 40^\circ \cos 40^\circ + \cos^2 40^\circ) + (\sin^2 50^\circ - 2 \sin 50^\circ \cos 50^\circ + \cos^2 50^\circ)$

$= (\sin^2 40^\circ + \cos^2 40^\circ) + (\sin^2 50^\circ + \cos^2 50^\circ) + 2 \sin 40^\circ \cos 40^\circ - 2 \sin 50^\circ \cos 50^\circ$

$= 1 + 1 + 2 \sin 40^\circ \cos 40^\circ - 2 \cos 40^\circ \sin 40^\circ$

$= 2$

(3)

$\sin 35^\circ \cos 35^\circ (\tan 35^\circ + \tan 55^\circ)$

$= \sin 35^\circ \cos 35^\circ (\frac{\sin 35^\circ}{\cos 35^\circ} + \frac{\sin 55^\circ}{\cos 55^\circ})$

$= \sin 35^\circ \cos 35^\circ (\frac{\sin 35^\circ}{\cos 35^\circ} + \frac{\cos 35^\circ}{\sin 35^\circ})$

$= \sin 35^\circ \cos 35^\circ (\frac{\sin^2 35^\circ + \cos^2 35^\circ}{\cos 35^\circ \sin 35^\circ})$

$= \sin 35^\circ \cos 35^\circ (\frac{1}{\cos 35^\circ \sin 35^\circ})$

$= 1$

3. 最終的な答え

(1) 1

(2) 2

(3) 1