Question

Prove: ⭐✨

$1+sinx = {sin(x/2) +cos(x/2)}^2$​

Answer 1

$\red {\sf {\large Identities \: \: Used }}$

$1) \: \: \: \: \: \sf { \sin }^{2} \alpha + {cos}^{2} \alpha = 1 \\ \\ \sf 2) \: \: \: \: \: sin2 \alpha = 2 \sin \alpha \cos\alpha$

Let's Move Towards The Question Now

$\red {\sf {\large To \: \: Prove }}$

$\sf1 + sin \: x = {(sin {} \dfrac{x}{2} + {cos} \dfrac{x}{2} ) } {}^{2}$

RHS

$\sf{(sin {}^{} \dfrac{x}{2} + {cos} \dfrac{x}{2} ) {}^{2} } {}\\ \\ = \sf {sin}^{2} \dfrac{x}{2} + {cos}^{2} \dfrac{x}{2} + 2sin \frac{x}{2} cos \dfrac{x}{2} \\ \\ \sf = 1 + sin \: x = LHS$

Hence Proved

Answer 2

Write sinx=2sinx/2cosx/2

Write sinx=2sinx/2cosx/2 And write 1 = sin2x/2+cos2x/2

Write sinx=2sinx/2cosx/2 And write 1 = sin2x/2+cos2x/2 So LHS becomes:

Write sinx=2sinx/2cosx/2 And write 1 = sin2x/2+cos2x/2 So LHS becomes:sin2x/2+cos2x/2+2sinx/2cosx/2 = [math](sin x/2 + cos x/2)^2]/math] = RHS === proved.