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Prove that: (sin7A+sin5A+sin9A+sin3A)/...

Prove that:
`(sin7A+sin5A+sin9A+sin3A)/(cos3A+cos5A+cos7A+cos9A)= tan6A`

Text Solution

Verified by Experts

LHS `((sin8A-sin4A)-(sin7A-sin5A))/((cos4A-cos8A)-(cos5A-cos7A))`
`(2cos((8A+4A)/(2)).sin((8A-4A)/(2))-2cos(7A+5A)/(2)sin(7A-5A)/(2))/(2sin(4A+8A)/(2).sin(8A-4A)/(2))`
`(2[cos6Asin2A-cos6AsinA])/(2[sin6Asin2A-sin6AsinA])`
`=(cos6A(sin2A-sinA))/(sin6A(sin2A-sinA))`
`=(cos6A)/(cos6A)=cot6A`= RHS Hence Proved.
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