Question

1+cosA/sinA=cotA/2 proof please

Answer 1

We have to rove that:

$\dfrac{1+\cos A}{\sin A}=\cot \dfrac{A}{2} .$

Solution:

L.H.S. = $\dfrac{1+\cos A}{\sin A}$

Using the trigonometric identities:

$2\cos^2 \dfrac{A}{2} =1+\cos A$

and $\sin A=2\sin \dfrac{A}{2}\cos \dfrac{A}{2}$

= $\dfrac{2\cos^2 \dfrac{A}{2}}{2\sin \dfrac{A}{2}\cos \dfrac{A}{2}}$

= $\dfrac{\cos \dfrac{A}{2}}{\sin \dfrac{A}{2}}$

= $\cot \dfrac{A}{2}$

= R.H.S., proved.

Thus, $\dfrac{1+\cos A}{\sin A}=\cot \dfrac{A}{2},$ proved.

Answer 2

Given:

1+cosA/sinA=cotA/2

To find:

Prove that, 1+cosA/sinA=cotA/2

Solution:

From given, we have,

(1 + cosA )/ sinA = cotA/2

Let us consider the LHS part of the above equation.

so, we have,

LHS:

(1 + cosA )/ sinA

now we will make use of 2 trigonometric properties.

1 + cosA = 2cos² A/2

sinA = 2sin A/2 cos A/2

substitute these values in the given equation. So, we get,

= (2cos² A/2) / (2sin A/2 cos A/2)

cancel out the common terms

= (cos A/2) / (sin A/2 )

we know that, cot A = cos A / sin A

RHS:

= cot A/2

Hence it is proved that, (1 + cosA )/ sinA = cotA/2