Question

Cos4theta+cos2theta/ cos4theta-cos2theta= _____

-cot3θcotθ
-2cot3θcotθ
cot6θcot2θ
-1

Answer 1

`\bf \textit{Sum to Product Identities} \\\\ cos(\alpha)+cos(\beta)=2cos\left(\cfrac{\alpha+\beta}{2}\right)cos\left(\cfrac{\alpha-\beta}{2}\right) \\\\\\ cos(\alpha)-cos(\beta)=-2sin\left(\cfrac{\alpha+\beta}{2}\right)sin\left(\cfrac{\alpha-\beta}{2}\right) \\\\[-0.35em] \rule{34em}{0.25pt}`

`\bf \cfrac{cos(4\theta )+cos(2\theta )}{cos(4\theta )-cos(2\theta )}\implies \cfrac{2cos\left( (4\theta +2\theta )/(2) \right)cos\left( (4\theta -2\theta )/(2) \right)}{-2sin\left( (4\theta +2\theta )/(2) \right)sin\left( (4\theta -2\theta )/(2) \right)} \implies \cfrac{cos\left( (6\theta )/(2) \right)cos\left( (2\theta )/(2) \right)}{-sin\left( (6\theta )/(2) \right)sin\left( (2\theta )/(2) \right)}`

`\bf \cfrac{cos(3\theta )cos(\theta )}{-sin(3\theta )sin(\theta )}\implies -\cfrac{cos(3\theta )}{sin(3\theta )}\cdot \cfrac{cos(\theta )}{sin(\theta )}\implies -cot(3\theta )cot(\theta )`