# 1-sinx Formula, Proof | 1+sinx Identity

The 1-sinx formula is given by 1-sinx= [cos(x/2) -sin(x/2)]2. The 1 plus sinx idenity is given as follows:

$1+\sin x =(\cos \frac{x}{2}+\sin \frac{x}{2})^2$.

Let us now find the formula of 1-sinx and 1+sinx.

## Formula of 1-sinx

Answer: 1-sinx= $(\cos \frac{x}{2}-\sin \frac{x}{2})^2$.

Explanation:

To find the formula of 1-sinx, we will use the following two trigonometric identities:

1. cos2θ +sin2θ = 1
2. sin2θ = 2sinθ cosθ

So the given expression 1-sinx can be written as

1- sinx

= cos2 $\frac{x}{2}$ +sin2 $\frac{x}{2}$ – 2sin $\frac{x}{2}$ cos $\frac{x}{2}$

= $(\cos \frac{x}{2}-\sin \frac{x}{2})^2$, here we have used the formula: a2+b2 -2ab=(a-b)2.

So the formula of 1-sinx is equal to [cos(x/2) -sin(x/2)]2.

Also Read: 1-cosx Formula, Identity with Proof

Sin3x formula in terms of sinx

sin(a+b) sin(a-b) Formula

## Formula of 1+sinx

Answer: 1+sinx is equal to $(\cos \frac{x}{2}+\sin \frac{x}{2})^2$.

Explanation:

As before, we have that

1+ sinx

= cos2 $\frac{x}{2}$ +sin2 $\frac{x}{2}$ + 2sin $\frac{x}{2}$ cos $\frac{x}{2}$

= $(\cos \frac{x}{2}+\sin \frac{x}{2})^2$, using the formula: a2+b2 +2ab=(a+b)2.

Therefore, the formula of 1+sinx is equal to [cos(x/2) + sin(x/2)]2.