Question

pls help pls pls pls​

Answer 1

We know that :

${\bigstar\;\;\boxed{\mathsf{SinA + SinB = 2.Sin\bigg(\dfrac{A + B}{2}\bigg).Cos\bigg(\dfrac{A - B}{2}\bigg)}}}$

${\bigstar\;\;\boxed{\mathsf{CosA + CosB = 2.Cos\bigg(\dfrac{A + B}{2}\bigg).Cos\bigg(\dfrac{A - B}{2}\bigg)}}}$

$\mathsf{Given :\;\;\dfrac{Sin3x + Sinx}{Cos3x + Cosx}}$

$\mathsf{\implies \dfrac{2.Sin\bigg(\dfrac{3x + x}{2}\bigg).Cos\bigg(\dfrac{3x - x}{2}\bigg)}{2.Cos\bigg(\dfrac{3x + x}{2}\bigg).Cos\bigg(\dfrac{3x - x}{2}\bigg)}}$

$\mathsf{\implies \dfrac{Sin\bigg(\dfrac{3x + x}{2}\bigg)}{Cos\bigg(\dfrac{3x + x}{2}\bigg)}}$

$\mathsf{\implies \dfrac{Sin\bigg(\dfrac{4x}{2}\bigg)}{Cos\bigg(\dfrac{4x}{2}\bigg)}}$

$\mathsf{\implies \dfrac{Sin(2x)}{Cos(2x)}}$

$\bigstar\;\;\textsf{We know that : \boxed{\mathsf{\dfrac{Sin\theta}{Cos\theta} = Tan\theta}}}$

$\mathsf{\implies Tan(2x)}$