SOLUTION: Use the sum-to-product and product-to-sum identities to prove: sin 8θ-sin 10θ = cot 9θ(cos10θ-cos8θ)

Algebra ->  Trigonometry-basics -> SOLUTION: Use the sum-to-product and product-to-sum identities to prove: sin 8θ-sin 10θ = cot 9θ(cos10θ-cos8θ)      Log On


   

Question 1161416: Use the sum-to-product and product-to-sum identities to prove:
sin 8θ-sin 10θ = cot 9θ(cos10θ-cos8θ)
Answer by Edwin McCravy(20060) About Me  (Show Source):
You can put this solution on YOUR website!
sin%288theta%29-sin%2810theta%29+=+cot%289theta%29%28cos%2810theta%29-cos%288theta%29%29

We work with the left side only first:

sin%288theta%29-sin%2810theta%29

Rewrite 8θ as 9θ-θ and 10θ as 9θ+θ

sin%289theta-theta%29-sin%289theta%2Btheta%29







-2cos%289theta%29sin%28theta%29

Now we work with the right side only:

cot%289theta%29%28cos%2810theta%29-cos%288theta%29%29

Rewrite 8θ as 9θ-θ and 10θ as 9θ+θ

cot%289theta%29%28cos%289theta%2Btheta%29-cos%289theta-theta%29%29







cot%289theta%29%28-2sin%289theta%29sin%28theta%29%5E%22%22%29+

%28cos%289theta%29%2Fsin%289theta%29%29%28-2sin%289theta%29sin%28theta%29%5E%22%22%29+



-2cos%289theta%29sin%28theta%29

Both sides work down the same expression.

Edwin