Question

Prove that cos³A+sin³A/cosA+sinA=1-cosA×sinA

Answer 1

Answer: Step-by-step explanation:

L.H.S=\frac{ COS^{3}A+ SIN^{3} A }{COS A+SIN A} + \frac{ COS^{3}A- SIN^{3}A  }{COS A-SIN A}  

L.H.S= \frac{(COS A-SIN A)( COS^{3}A + SIN^{3}A )+(COS A+SIN A)( COS^{3}A - SIN^{3}A )}{(COS A+SIN A)(COS A-SIN A)}  

L.H.S= \frac{2 COS^{4}A-2 SIN^{4}A }{ COS^{2}A- SIN^{2}A }  

L.H.S=2 \frac{ ( COS^{2}A) ^{2}- ( SIN^{2}A )^{2}  }{COS^{2}A- SIN^{2}A}  

L.H.S=2 \frac{ \frac{ (COS 2A+1)^{2} }{ 2^{2} }- \frac{(1-COS 2A)^{2}}{ 2^{2} }  }{COS^{2}A- SIN^{2}A}  

L.H.S= \frac{ \frac{ COS^{2}2A+1+2COS2A-1- COS^{2}2A+2COS2A  }{2} }{COS2A}  

L.H.S= \frac{ \frac{4COS2A}{2} }{COS2A}  

L.H.S= \frac{2COS2A}{COS2A}  

L.H.S=2

L.H.S=R.H.S

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