Question

prove L.H.S = R.H.S​

Answer 1

Answer:

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Answer 2

Answer:

(1 + cos ∅ + sin ∅)/(1 + cos ∅ - sin ∅) = (1 + sin ∅)/cos ∅

Proof:

LHS :

( 1 + cos ∅ + sin ∅)/( 1 + cos ∅ - sin ∅)

Multiplying the numerator and denominator with cos ∅ :

⇒ [cos ∅( 1 + cos ∅ + sin ∅)]/[cos ∅( 1 + cos ∅ - sin ∅)]

⇒ (cos ∅ + cos² ∅ + cos ∅ sin ∅)/(cos ∅ + cos² ∅ - cos ∅ sin ∅)

Taking cos ∅ common :

⇒ [cos ∅ (1 + sin ∅) + cos² ∅]/[cos ∅(1 + cos ∅ - sin ∅)]

Using identity - cos² ∅ = 1 - sin² ∅ :

⇒ [cos ∅ (1 + sin ∅) + 1 - sin² ∅]/[cos ∅(1 + cos ∅ - sin ∅)]

⇒ [cos ∅ (1 + sin ∅) + (1)² - (sin ∅)²]/[cos ∅(1 + cos ∅ - sin ∅)]

⇒ [cos ∅ (1 + sin ∅) + (1 + sin ∅)(1 - sin ∅)]/[cos ∅(1 + cos ∅ - sin ∅)]

Taking 1 + sin ∅ as common in numerator :

⇒ [1 + sin ∅(cos ∅ + 1 - sin ∅)]/[cos ∅(1 + cos ∅ - sin ∅)]

⇒ [1 + sin ∅(1 + cos ∅ - sin ∅)]/[cos ∅(1 + cos ∅ - sin ∅)]

⇒ (1 + sin ∅)/cos

Hence Proved.