cos θ + sin θ = √2 cos θ, at cos θ - sin θ= ?
(1) √2 tan θ
(2) - √2 cos θ
(3) - √2 sin θ
(4) √2 sin θ
Answers
Answered by
54
Answer
√2 sin∅
Explanation
Given
cos∅ + sin∅ = √2cos∅
Squaring both sides,
(cos∅ + sin∅)² = (√2cos∅)²
⇒ cos²∅ + sin²∅ + 2sin∅cos∅ = 2 cos²∅
⇒ 2sin∅cos∅ = 2cos²∅ - cos²∅ - sin²∅
⇒ 2sin∅cos∅ = cos²∅ - sin²∅
Now, using a² - b² = (a + b)(a - b)
⇒ 2sin∅cos∅ = (cos∅ + sin∅)(cos∅ - sin∅)
Now, we know that (cos∅ + sin∅) = √2cos∅
⇒ 2sin∅cos∅ = (√2cos∅)(cos∅ - sin∅)
⇒ cos∅ - sin∅ = 2sin∅cos∅/√2cos∅
⇒ cos∅ - sin∅ = √2sin∅
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