Question

prove that sin9•+sin•+sin5•+sin7•/cos9•+cos•+cos5•+cos7•=tan4•​

Answer 1

Answer:

Proved

Step-by-step explanation:

Prove that sin9•+sin•+sin5•+sin7•/cos9•+cos•+cos5•+cos7•=tan4•​

$\frac{Sin9x + Sinx + Sin5x + Sin7x}{Cos9x + Cosx + Cos5x + Cos7x} = Tan4x$

Correction in question

$\frac{Sin3x + Sinx + Sin5x + Sin7x}{Cos3x + Cosx + Cos5x + Cos7x} = Tan4x$

=> $\frac{Sin3x + Sin5x + Sinx + Sin7x}{Cos3x + Cos5x + Cosx + Cos7x} = Tan4x$

Using Sinx + Siny = 2Sin((x+y)/2)Cos((x-y)/2)

         Cosx + Cosy = 2Cos((x+y)/2)Cos((x-y)/2)

=> $\frac{2Sin4xCosx + 2Sin4xCos3x}{2Cos4xCosx + 2Cos4xCos3x} = Tan4x$

=> $\frac{2Sin4x(Cosx + Cos3x)}{2Cos4x(Cosx + Cos3x)} = Tan4x$

cancelling 2(Cosx + Cos3x)

=> $\frac{Sin4x}{Cos4x} = Tan4x$

=> Tan4x = Tan4x

QED