Question

If cosФ+sinФ=√2cosФ.Show that cosФ-sinФ=√2sinФ[Don't spam]

Answer 1

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Answer 2

Given

✎ cosФ+sinФ=√2cosФ

To show

✎ cosФ-sinФ=√2sinФ

Proof

Let's start with what we have been given :

⇒ cosθ + sinθ = √2cosθ

⇒ sinθ = √2cosθ - cosθ

⇒ sinθ = (√2 - 1)cosθ

⇒ (√2 + 1)sinθ = (√2 + 1)(√2 - 1)cosθ [Multiplying (√2 + 1) on both sides]

⇒ √2sinθ + sinθ = [(√2)² - 1]cosθ

⇒ √2sinθ + sinθ = (2 - 1)cosθ

⇒ √2sinθ = cosθ - sinθ

⇒ cosθ - sinθ = √2sinθ [Showed]

∴ LHS = RHS [Showed]

Some more trigonometric identities :

✎ sin²θ + cos²θ = 1

✎ tanθ = sinθ/cosθ

✎ cotθ = cosθ/sinθ

✎ sec²θ - tan²θ = 1

✎ sinθ = 1/cscθ

✎ cosθ = 1/secθ

✎ tanθ = 1/cotθ