Prove the following trigonometric identities.
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Answer with Step-by-step explanation:
Given: cosA/(1 − tanA) + sinA /(1 − cotA) = sin A + cosA
LHS : cosA/(1 − tanA) + sinA /(1 − cotA)
= cosA/(1 − sinA/cosA) + sinA/(1 − cosA/sinA)
[By using the identity, cotθ = cosθ/sinθ , tanθ = sinθ/cosθ ]
= cosA/[(cosA− sinA)/cosA)] + sinA/[(sinA − cosA)/sinA]
[By taking LCM]
= cosA × cosA /[(cosA− sinA)] + sinA × sinA / [(sinA − cosA)]
= cos²A/(cosA−sinA) − sin² A /(cosA − sinA)
= (cos²A − sin²A)/(cosA−sinA)
[By taking LCM]
= [(cosA + sinA)(cosA − sinA)] / (cosA − sinA)
[By using identity , a² - b² = (a + b) (a - b)]
= cos A + sin A
cosA / (1 − tanA) + sinA /(1 − cotA) = sin A + cosA
L.H.S = R.H.S
Hence Proved..
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