Question

Hey friends...
Prove that sin A / 1+ cos A = cosec A - cot A

Answer 1

Hey !!!

Here is ur solution ;-

from LHS

sinA/1 + cosA

multiplying by (1 - cosA) on numerator and denominator

sinA ___(1 - cosA)
----------×
1+cosA_(1-cosA)

sinA - sinA×cosA
-----------------------
1² - cos²A

sinA - sinA×cosA
------------------------
sin²A



sin(1-cosA)
---------------
sin²A

1 - cosA
------------
sinA

=> 1/sinA -cosA/sinA

=> cosecA - cotA RHS prooved

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Hope it helps you !!

@Rajukumar111

Answer 2

Solution :

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Given :

To prove that :

⇒ $\frac{Sin A}{1 + Cos A} = Cosec A - Cot A$

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Proof :

LHS = $\frac{Sin A}{1 + Cos A}$

By multiplying with 1 - Cos A both the sides,

We get,

⇒ $\frac{Sin A}{1 + Cos A} ( \frac{1 - Cos A}{1 - Cos A} )$

⇒ $\frac{Sin A (1 - Cos A)}{(1 + Cos A)(1 - Cos A)}$

The denominator is in the form,

⇒ (a + b)(a - b),.

Hence,

We can use this identity : (a - b)(a + b) = a² - b²


⇒ $\frac{Sin A (1 - Cos A)}{1^2 - Cos^{2} A }$

⇒ $\frac{Sin A (1 - Cos A)}{1 - Cos^{2} A}$


We know that,

⇒ Sin² A + Cos² A = 1

∴ Sin²A = 1 - Cos² A

⇒ $\frac{Sin A (1 - Cos A)}{ Sin^2 A }$

⇒ $\frac{Sin A (1 - Cos A)}{(Sin A)(Sin A)}$

⇒ $\frac{1 - Cos A}{Sin A}$

⇒ $\frac{1}{Sin A} - \frac{Cos A}{Sin A}$

⇒ $Cosec A - Cot A$

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                                            Hope it Helps !!