Bank Soal Matematika SMA Fungsi Trigonometri dan Bilangan Real

Perlu diingat identitas-identitas pada trigonometri berikut:

$\sin^2\theta+\cos^2\theta=\cos^2\theta+\sin^2\theta=1$

Berdasarkan identitas-identitas pada trigonometri di atas diperoleh:

$\frac{\cos^4\theta-\sin^4\theta}{\sin\theta-\cos\theta}$

$=\frac{\left(\cos^2\theta+\sin^2\theta\right)\left(\cos^2\theta-\sin^2\theta\right)}{\sin\theta-\cos\theta}$; ingat bahwa $a^4-b^4=\left(a^2+b^2\right)\left(a^2-b^2\right)$

$=\frac{\left(1\right)\left(\cos^2\theta-\sin^2\theta\right)}{\sin\theta-\cos\theta}$

$=\frac{\left(\cos^2\theta-\sin^2\theta\right)}{\sin\theta-\cos\theta}$

$=\frac{\left(\cos\theta+\sin\theta\right)\left(\cos\theta-\sin\theta\right)}{\sin\theta-\cos\theta}$; ingat bahwa $a^2-b^2=\left(a+b\right)\left(a-b\right)$

$=\frac{\left(\cos\theta-\sin\theta\right)\left(\cos\theta+\sin\theta\right)}{\sin\theta-\cos\theta}$

$=\frac{-\left(\sin\theta-\cos\theta\right)\left(\cos\theta+\sin\theta\right)}{\sin\theta-\cos\theta}$

$=-\left(\cos\theta+\sin\theta\right)$

$=-\cos\theta-\sin\theta$

$=-\sin\theta-\cos\theta$

Jadi, nilai dari $\frac{\cos^4\theta-\sin^4\theta}{\sin\theta-\cos\theta}$ adalah $-\sin\theta-\cos\theta$.