prove that sin A/1-cosA=cosec A+cotA
Answers
Step-by-step explanation:
Solution :
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Given :
To prove that :
⇒ \frac{Sin A}{1 + Cos A} = Cosec A - Cot A
1+CosA
SinA
=CosecA−CotA
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Proof :
LHS = \frac{Sin A}{1 + Cos A}
1+CosA
SinA
By multiplying with 1 - Cos A both the sides,
We get,
⇒ \frac{Sin A}{1 + Cos A} ( \frac{1 - Cos A}{1 - Cos A} )
1+CosA
SinA
(
1−CosA
1−CosA
)
⇒ \frac{Sin A (1 - Cos A)}{(1 + Cos A)(1 - Cos A)}
(1+CosA)(1−CosA)
SinA(1−CosA)
The denominator is in the form,
⇒ (a + b)(a - b),.
Hence,
We can use this identity : (a - b)(a + b) = a² - b²
⇒ \frac{Sin A (1 - Cos A)}{1^2 - Cos^{2} A }
1
2
−Cos
2
A
SinA(1−CosA)
⇒ \frac{Sin A (1 - Cos A)}{1 - Cos^{2} A}
1−Cos
2
A
SinA(1−CosA)
We know that,
⇒ Sin² A + Cos² A = 1
∴ Sin²A = 1 - Cos² A
⇒ \frac{Sin A (1 - Cos A)}{ Sin^2 A }
Sin
2
A
SinA(1−CosA)
⇒ \frac{Sin A (1 - Cos A)}{(Sin A)(Sin A)}
(SinA)(SinA)
SinA(1−CosA)
⇒ \frac{1 - Cos A}{Sin A}
SinA
1−CosA
⇒ \frac{1}{Sin A} - \frac{Cos A}{Sin A}
SinA
1
−
SinA
CosA
⇒ Cosec A - Cot ACosecA−CotA
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Hope it Helps !!
Answer:
L.H.S.= sin A/(1-cos A) . Multiply and divide with (1+cos A)
= sin A(1+cos A)/(1-cos A)(1+cos A)
= sin A(1+cos A)/(1-cos^2 A)
= sin A(1+cos A)/sin^2 A
=(1+cos A)/sin A
= 1/sinA + cosA/sinA
= cosecA+cotA= RHS