Sin A. Cos A. Sec A. Cosec A divided
Tan A. Cot =1
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Prove that sin A/(sec A + tan A – 1) + cos A/(cosec A + cot A – 1) = 1
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asked Mar 30, 2019 in Class X Maths by priya12 (-12,631 points)
Prove that sin A/(sec A + tan A – 1) + cos A/(cosec A + cot A – 1) = 1
trigonometry
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answered Mar 30, 2019 by muskan15 (-3,443 points)
L.H.S.
= sin A/(sec A + tan A – 1) + cos A/(cosec A + cot A – 1)
= [sin A/(1/cos A + sin A/cos A) – 1] + [cos A/(1/sin A + cos A/sin A) – 1]
= [sin A/(1 + sin A – cos A)/cos A] + [cos A/(1 + cos A – sin A)/sin A]
= sin A.cos A/1 + sin A – cos A + sin A.cos A/1 + cos A – sin A
= sin A. cos A (1 + cos A – sin A + 1 + sin A – cos A)/[1 + (sin A – cos A)][1 – (sin A – cos A)]
= 2 sin A.cos A/(1)2 – (sin A – cos A)2
= 2 sin A.cos A/(1 – (sin2 A + cos2 A – 2 sin A.cos A)
= 2 sin A.cos A/1 – 1 + 2 sin A.cos A
= 2/2 = 1 = R.H.S.
Hence proved.
Answer:
L.H.S
Sin A × Cos A × Sec A × Cosec A / tan A × Cot A = 1
Firstly we will solve the upper side or the numerator.
Sin A × Cos A × 1/Cos A × 1/ Sin A = 1. ( here Sin A is divided by 1/ Sin A and Cos A is divided by 1/ Cos A )
Now, we will solve the denominator.
Tan A × 1/Tan A = 1. ( here tan A is divided by 1/ Tan A)
numerator/ Denominator
i.e 1/1
R.H.S = 1
SO, L.H.S = R.H.S
1 = 1
hence proved.