Question

prove
that
(1+ sino + coso
$(1 + \sin(x) + \cos(x) ) = (1 + \sin(x) + (1 + \cos(x) )$
) =(1+Sine)(1+coso)​

Answer 1

Step-by-step explanation:

To Prove : (1 + sinθ + cosθ)² = 2(1 + sinθ)(1 + cosθ)

Proof : L.H.S. = (1 + sinθ + cosθ)²

• Identity : (a + b + c)² = a² + b² + c² + 2ab + 2bc + 2ca

Here, a = 1, b = sinθ, c = cosθ

→ (1)² + (sinθ)² + (cosθ)² + 2(1)(sinθ) + 2(sinθ)(cosθ) + 2(cosθ)(1)

→ 1 + sin²θ + cos²θ + 2sinθ + 2sinθcosθ + 2cosθ

• Identity : sin²θ + cos²θ = 1

→ 1 + 1 + 2sinθ + 2sinθcosθ + 2cosθ

→ 2 + 2sinθ + 2sinθcosθ + 2cosθ

Taking common terms out.

→ 2(1 + sinθ) + 2cosθ(sinθ + 1)

Rearranging the terms.

→ 2(1 + sinθ) + 2cosθ(1 + sinθ)

→ (2 + 2cosθ)(1 + sinθ)

Taking common terms out again.

→ 2(1 + cosθ)(1 + sinθ)

= R.H.S.

Hence, proved !!