Question

2cos4x+1/2cosx+1 = (2cosx-1) (2cos2x-1)​

Answer 1

Answer:

Step-by-step explanation:

Consider the provided equation.

$\frac{2cos4x+1}{2cosx+1} =(2cosx-1) (2cos2x-1)$

Now consider the right hand side. Multiply the right hand side with $\frac{(2cosx+1)}{(2cosx+1)}$

$(2cosx-1)(2cos2x-1)\times \frac{2cosx+1}{2cosx+1}$

$\frac{(4cos^2x-1)(2cos2x-1)}{2cosx+1}$

Now use the identity: $\cos^2x=\dfrac{1+\cos2x}2$

By using the above identity:

$\frac {(\frac{4(1+cos2x)}{2}-1)(2cos2x-1)}{2cosx+1}$

$\frac {(2+2cos2x-1)(2cos2x-1)}{2cosx+1}$

$\frac {(2cos2x+1)(2cos2x-1)}{2cosx+1}$

$\frac{4cos^22x-1}{2cosx+1}$

Again use the identity: $\cos^2x=\dfrac{1+\cos2x}2$

$\frac{\frac{4(1+cos4x)}{2}-1}{2cosx+1}$

$\frac{2+2cos4x-1}{2cosx+1}$

$\frac{2cos4x+1}{2cosx+1}$

Hence, proved.