Question

sinA/1-cosA=cosecA+cotA

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Answer 1

Step-by-step explanation:

Answer and Explanation:

To prove : \frac{\sin A}{1+\cos A}=\csc A-\cot A

1+cosA

sinA

=cscA−cotA

Proof :

Taking LHS,

\frac{\sin A}{1+\cos A}

1+cosA

sinA

Rationalize,

=\frac{\sin A}{1+\cos A}\times \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)

=\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)}{\sin^2 A}=

sin

2

A

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

=\csc A-\tan A=cscA−tanA

= RHS

So, LHS=RHS hence proved.

Answer 2

sin A

1 + cos A

sin A 1 LHS = cos A

1 + cos A 1 - cos A

sin A(1 - cos A)

1 - cos² A

1 cos A

sin A

1

cos A

sin A sin A

= cosecA - cotA RHS