Question

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Answer 1

ANSWER

cotAcosA/cotA+cosA


=[(cosA/sinA)cosA]/[(cosA/sinA)+cosA]


=(cos²A/sinA)/[(cosA+sinAcosA)/sinA]


=cos²A/cosA(1+sinA)


=cosA/(1+sinA)


=[cosA(1-sinA)]/[(1+sinA)(1-sinA)]


 [multiplying the numerator and the denominator with (1-sinA)]


=[cosA(1-sinA)]/(1-sin²A)


=(cosA-cosAsinA)/cos²A


=[(cosA-cosAsinA)/sinA]/(cos²A/sinA)


[dividing the numerator and the denominator by sinA]

=[(cosA/sinA)-(cosAsinA/sinA)]/(cosA/sinA)cosA



=(cotA-cosA)/cotAcosA (Proved)


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Answer 2

L.H.S

cot A cos A / cot A + cos A

Use cot A = cos A / sin A

⇒ ( ( cos A / sin A ) cos A ) / ( ( cos A / sin A)+ cos A )

⇒ ( cos² A/sin A) / ( ( cos A + sin A cos A) / sin A )

Cancel sin A

⇒ cos²A / cos A ( 1 + sin A )

R.H.S

⇒ ( cot A - cos A ) / ( cot A cos A )

⇒ ( cos A / sin A - cos A ) / ( cos² A / sin A )

⇒ ( cos A - sin A cos A ) / sin A / cos² A / sin A

⇒ ( cos A - sin A cos A ) / cos² A

⇒ cos A ( 1 - sin A ) / cos² A

⇒ ( 1 - sin A ) / cos A

⇒ ( 1  - sin A )( 1 + sin A ) / cos A ( 1 + sin A )

⇒ ( 1 - sin²A ) / cos A ( 1  +  sin A )

⇒ cos² A / cos A ( 1 + sin A )

L.H.S = R.H.S [ Proved ]