If α+α={2}α then what is the value of α-α?

Correct Answer is : 2sin⁡α{2}α2sinα

Correct Answer is : 2sin⁡α{2}α2sinα

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Trigonometry Practice Test 2

Question : 5 of 22

Marks: +1, -0

\cos\alpha+\sin\alpha=\sqrt{2}\cos\alpha

\cos\alpha-\sin\alpha?

Solution:

Concept:

Use the given equation to express

\sin\alpha

in terms of

\cos\alpha

and compare with each option.

Explanation:

Given:

\cos\alpha+\sin\alpha=\sqrt{2}\cos\alpha

Rearrange:

\sin\alpha=(\sqrt{2}-1)\cos\alpha

Now,

\cos\alpha-\sin\alpha=\cos\alpha-(\sqrt{2}-1)\cos\alpha

Simplify:

\cos\alpha-\sin\alpha=(2-\sqrt{2})\cos\alpha

Next, compute

\sqrt{2}\sin\alpha=\sqrt{2}(\sqrt{2}-1)\cos\alpha

Simplify:

\sqrt{2}\sin\alpha=(2-\sqrt{2})\cos\alpha

Both expressions are equal, so

\cos\alpha-\sin\alpha=\sqrt{2}\sin\alpha

Since

\sqrt{2}-1>0

\sin\alpha

and

\cos\alpha

have the same sign, so this equality is valid without sign ambiguity.

Answer:

\cos\alpha-\sin\alpha=\sqrt{2}\sin\alpha

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