Question

How to prove that
$\frac{sinx}{cos x +1 } = tan \frac{x}{2}$

Answer 1

Step-by-step explanation:

Given :-

Sin x / ( Cos x +1)

To find :-

Prove that Sin x / ( Cos x +1) = Tan x/2

Solution :-

On taking LHS

Sin x / ( Cos x +1)

We know that

Sin x = 2 Sin (x/2) Cos (x/2)

and

Cos x = 2 Cos² (x/2) - 1

Using these identities in the given expression then

=>[2 Sin (x/2) Cos (x/2)]/[ 2 Cos² (x/2) -1+1]

=> [2 Sin (x/2) Cos (x/2)]/[2 Cos² (x/2)]

=> [2Sin (x/2)Cos(x/2)]/[2Cos(x/2)Cos(x/2)]

On cancelling 2 Cos (x/2) in both numerator and denominator then

=> Sin (x/2) / Cos (x/2)

=> Tan (x/2)

=> RHS

=> LHS = RHS

Hence, Proved.

Used formulae:-

→ Sin x = 2 Sin (x/2) Cos (x/2)

→ Cos x = 2 Cos² (x/2) - 1

→ Tan x = Sin x / Cos x