Question

Express each of the following as a product: `sin4theta+sin2theta` (ii) `sin6theta-sin2theta` (iii) `cos4theta+cos8theta` (iii) `cos6theta-cos8theta`

Answer 1

We will use the below rules,
`sinA + sinB = 2sin((A+B)/2)cos((A-B)/2)`
`sinA-sinB = 2sin((A-B)/2)cos((A+B)/2)`
`cosA + cosB = 2cos((A+B)/2)cos((A-B)/2)`
`cosA - cosB = 2sin((A+B)/2)sin((B-A)/2)`
Now, (i) `sin4theta+sin2theta = 2sin((4theta+2theta)/2)cos((4theta-2theta)/2)`
`= 2sin3thetacostheta`

(ii)`sin6theta-sin2theta = 2cos((6theta+2theta)/2)sin((6theta-2theta)/2)`
`= 2cos4thetasin2theta`

(iii) `cos4theta+cos8theta = 2cos((4theta+8theta)/2)cos((4theta-8theta)/2)`
`= 2cos6thetacos(-2theta)`
`= 2cos6thetacos2theta...[As cos(-theta) = costheta]`

(iv) `cos6theta-cos8theta = 2sin((6theta+8theta)/2)cos((8theta-6theta)/2)`
`= 2sin7thetasintheta`