Question

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Answer 1

Question :

If √3tan ∅ = 3sin ∅, prove that ( sin² ∅ - cos² ∅ ) = 1/3.

[ Note : theta is written as ∅ ]

Answer :

Given

√3tan ∅ = 3sin ∅

Since tan ∅ = sin ∅ / cos ∅

⇒ √3 × ( sin ∅ / cos ∅ ) = 3sin ∅

⇒ sin ∅ / cos ∅. sin ∅ = 3 / √3

⇒ 1 / cos ∅ = 3 / √3

⇒ cos ∅ = √3 / 3

It can be written as

⇒ cos ∅ = √3 /  ( √3 ×√3 )

⇒ cos ∅ = 1 / √3

Squaring on both sides

⇒ cos² ∅ = ( 1 / √3 )²

⇒ cos² ∅ = 1 / 3 → ( 1 )

Now, let's  find the value of sin² ∅ using trignometric identity sin² ∅ = 1 - cos² ∅

⇒ sin² ∅ = 1 - 1/3

⇒ sin² ∅ = ( 3 - 1 )/3

⇒ sin² ∅ = 2/3 → ( 2 )

Subtracting ( 1 ) from ( 2 ) we get,

⇒ sin² ∅ - cos² ∅ = 2/3 - 1/3

⇒ sin² ∅ - cos² ∅ = ( 2 - 1 )/3

⇒ sin² ∅ - cos² ∅ = 1/3

Hence proved.

Answer 2

Answer:

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