Trigonometry
Trigonometric Identities
MMTS_Full_Test_02
Grade 12
Question:
If $\sin\theta+\cos\theta=\sqrt{2}\cos\theta$, find $\cos\theta-\sin\theta$
$\sqrt{2}\cos\theta$
$\sqrt{2}\sin\theta$
$-\sqrt{2}\cos\theta$
$-\sqrt{2}\sin\theta$
Step-by-Step Solution
Key Concept: $\sin\theta=(\sqrt{2}-1)\cos\theta$; compute $\cos\theta-\sin\theta$
From $\sin\theta+\cos\theta=\sqrt{2}\cos\theta$: $\sin\theta=(\sqrt{2}-1)\cos\theta$. $\cos\theta-\sin\theta=\cos\theta-(\sqrt{2}-1)\cos\theta=(2-\sqrt{2})\cos\theta=\sqrt{2}(\sqrt{2}-1)^{-1}\cdot\sin\theta=\sqrt{2}\sin\theta$.
Correct Answer: 2