SinA+sin3A+sin5A+sin7A/cosA+cos3A+cos5A+cos7A=2tan2A/1-tan^22A
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LHS=sinA−sin3A+sin5A−sin7AcosA−cos3A−cos5A+cos7A=(sin5A+sinA)−(sin7A+sin3A)(cos7A+cosA)−(cos5A+cos3A)=2sin5A+A2cos5A−A2−2sin7A+3A2cos7A−3A22cos7A+A2cos7A−A2−2cos5A+3A2cos5A−3A2=2sin3A cos2A−2sin5A cos2A2cos4A cos3A−2cos4A cosA=2cos2A(sin3A−sin5A)2cos4A(cos3A−cosA)=cos2A(sin5A−sin3A)cos4A(cosA−cos3A)=cos2A×2cos5A+3A2sin5A−3A2cos4A×2sinA+3A2sin3A−A2=cos2A×cos4A×sinAcos4A×sin2A×sinA=cos2Asin2A=cot2A=RHSHence, proved.
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