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Question

Prove the following identities (1-17)

1-sin A cos Acos A sec A-cosec A·sin2 A-cos2 Asin3 A+cos3 A=sin A

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Solution

LHS=1-sin A cos Acos A sec A-cosec A×sin2 A-cos2 Asin3 A+cos3 A =1-sin A cos Acos A 1cos A-1sin A×sin A2-cos A2sin A3+cos A3 =1-sin A cos Acos A sin A-cos Acos A sin A×sin A+cos Asin A-cos Asin A+cos Asin A2+cos A2-sin A cos A = sin A1-sin A cos A sin A-cos A×sin A+cos Asin A-cos Asin A+cos Asin A2+cos A2-sin A cos A = sin A1-sin A cos A 1×1×1sin2 A+cos2 A-sin A cos A =sin A1-sin A cos A×11-sinA cos A =sin A =RHSHence proved.

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