Question

to prove cos3a+sin3a/cosa-sina=1+2sin2a

Answer 1

cos3a = 4cosa - 3cos³a, sin3a = 3sina - 4sin³a,

substituting we get, 

(4cos³a - 3cosa + 3sina - 4sin³a)/cosa-sina 
= 4(cos³a - sin³a) - 3(cosa-sina) / cosa-sina
= 4(cos²a+sin²a+sinacosa) - 3
= 4 - 3 + 4sinacosa 
= 1 + 2sin2a. 

hence proved

Answer 2

Answer:

cos3a = 4cosa - 3cos³a, sin3a = 3sina - 4sin³a

LHS=(4cos³a - 3cosa + 3sina - 4sin³a)/cosa-sina 

= 4(cos³a - sin³a) - 3(cosa-sina) / cosa-sina

= 4(cos²a+sin²a+sinacosa) - 3

= 4 - 3 + 4sinacosa 

= 1 + 2sin2a. 

=RHS

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