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Prove that: (sin5A-sin3A)/(cos5A+cos3A)=...

Prove that: `(sin5A-sin3A)/(cos5A+cos3A)=tanA` `(sinA-sin3A)/(cosA+cos3A)=tan2A`

Text Solution

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(i) LHS `=(sin5A-sin3A)/(cos 5A+cos 3A)`
`=(2sin((5A-3A)/(2))cos((5A+3A)/(2)))/(2 cos ((5A+3A)/(2))cos((5A-3A)/(2)))`
`=(2sin Acos 4A)/(2 cos 4A cosA)=tan A=`RHS.
(ii) LHS`=(sin 3A+sinA)/(cos3A+cosA)`
`=(2sin((3A+A)/(2))cos((3A-A)/(2)))/(2cos((3A+A)/(2))cos((3A-A)/(2)))`
`=(sin2AcosA)/(cos 2Acos A)=tan 2A=`RHS.
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