Cos2 A/1-tanA + sin3A/sinA-cosA = 1+ sinAcosA Please answer this?

Question

#cos^2 A/(1-tanA) + sin^3A/(sinA-cosA) = 1+ sinAcosA#
Please Prove this.

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Answer 1 · 2018-02-20T10:11:20

Please refer to a Proof in the Explanation.

I hope that the Problem is :

#"Prove : "cos^2 A/(1-tanA) + sin^3A/(sinA-cosA) = 1+ sinAcosA.#

We have, #cos^2 A/(1-tanA) + sin^3A/(sinA-cosA)#,

#=cos^2A/(1-sinA/cosA)-sin^3A/(cosA-sinA)#,

#=cos^2A/{(cosA-sinA)/cosA}-sin^3A/(cosA-sinA)#,

#=cos^3A/(cosA-sinA)-sin^3A/(cosA-sinA)#,

#=(cos^3A-sin^3A)/(cosA-sinA)#,

#={cancel((cosA-sinA))(cos^2A+cosAsinA+sin^2A)}/cancel((cosA-sinA))..........(ast)#,

#=(cos^2A+sin^2A+sinAcosA)#,

#=1+sinAcosA#, as desired!

Note that while cancelling #(cosA-sinA)" at "(ast)#, we have

assumed that #(cosA-sinA)!=0, i.e., tanA!=1#.

This is quite admissible, otherwise the left member of the

Identity would not exist!