sinA sin2A +Sin3A Sin6A /cos2A SinA +Sin3A Cos6A
Answers
Answer:
Step-by-step explanation:
sin A sin 2A + sin 3A sin 6A/ sin A cos 2A + sin 3A cos 6A = tan 5A
We will use the following formulae in this solution--
(1) Sin A . Sin B = (1/2) [ Cos (A-B) - Cos (A+B) ]
(2) Sin A Cos B = (1/2) [ Sin (A+B) + Sin (A-B) ]
(3) Sin A + Sin B = 2 Sin (1/2) (A+B) . Cos (1/2)(A-B)
(4) Sin A - Cos B = 2 Sin (1/2) (A-B) . Cos (1/2)(A+B)
(5) Sin ( -x ) = - Sin x
(6) Cos ( - x ) = Cos x
Solution --
Numerator = (1/2) [ 2 SinA Sin 2A + 2 Sin 3A. Sin 6A ] --- Use formula (1)
=> (1/2) [ Cos (A-2A) - Cos (A+2A) + Cos (3A-6A) - Cos (3A+6A) ]
=> (1/2) [ Cos A - Cos 3A + Cos 3A - Cos 9A ]
=> (1/2) [ Cos A - Cos 9A ] .... Again use formula (1)
=> Sin 5A . Sin 4A
Denominator
= Sin A Cos 2A + Sin 3A.Cos 6A
=> (1/2) [ Sin (A+2A) + Sin (A-2A) + Sin ( 3A + 6A ) + Sin ( 3A - 6A ) ]
=> (1/2) [ Sin 3A - Sin A + Sin 9A - Sin 3A ]
=> (1/2) [ Sin 9A - Sin A ]
=> (1/2) [ 2 Cos 5A . Sin 4A ]
=> Cos 5A . Sin 4A
Hence the Left Hand Side of the given identity
= Numerator / Denominator
=> Sin 5A . Sin 4A / Cos 5A . Sin 4A
=> Tan 5A = RHS ………………. QED
Answer:
Step-by-step explanation:
sin A sin 2A + sin 3A sin 6A/ sin A cos 2A + sin 3A cos 6A = tan 5A
We will use the following formulae in this solution--
(1) Sin A . Sin B = (1/2) [ Cos (A-B) - Cos (A+B) ]
(2) Sin A Cos B = (1/2) [ Sin (A+B) + Sin (A-B) ]
(3) Sin A + Sin B = 2 Sin (1/2) (A+B) . Cos (1/2)(A-B)
(4) Sin A - Cos B = 2 Sin (1/2) (A-B) . Cos (1/2)(A+B)
(5) Sin ( -x ) = - Sin x
(6) Cos ( - x ) = Cos x
Solution --
Numerator = (1/2) [ 2 SinA Sin 2A + 2 Sin 3A. Sin 6A ] --- Use formula (1)
=> (1/2) [ Cos (A-2A) - Cos (A+2A) + Cos (3A-6A) - Cos (3A+6A) ]
=> (1/2) [ Cos A - Cos 3A + Cos 3A - Cos 9A ]
=> (1/2) [ Cos A - Cos 9A ] .... Again use formula (1)
=> Sin 5A . Sin 4A
Denominator
= Sin A Cos 2A + Sin 3A.Cos 6A
=> (1/2) [ Sin (A+2A) + Sin (A-2A) + Sin ( 3A + 6A ) + Sin ( 3A - 6A ) ]
=> (1/2) [ Sin 3A - Sin A + Sin 9A - Sin 3A ]
=> (1/2) [ Sin 9A - Sin A ]
=> (1/2) [ 2 Cos 5A . Sin 4A ]
=> Cos 5A . Sin 4A
Hence the Left Hand Side of the given identity
= Numerator / Denominator
=> Sin 5A . Sin 4A / Cos 5A . Sin 4A
=> Tan 5A = RHS ………………. QED