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student-nameKunal Pawar asked in Math

If cot Ѳ = 2n2 + 2mn / m2 + 2mn, then prove that cosec Ѳ = m2 + 2mn +2n2 / m2 +2mn

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student-nameAshutosh Verma answered this

in Math, Class

Answer :

we have Cot θ = 2n2 + 2mn m2 + 2mn

And

we know

Cot θ = AdjacentOpposite , So

Adjacent = 2n2 + 2mn And Opposite = m2 + 2mn

And we know

Hypotenuse 2 = Adjacent 2 + Opposite 2 , Substitute values we get

Hypotenuse 2 = ( 2n2 + 2mn ) 2 + ( m2 + 2mn ) 2

Hypotenuse 2 = ( 4n4 + 4m2n2 +8mn3 ) + ( m4 + 4m2n2 +4m3n )

Hypotenuse 2 = 4n4 + 4m2n2 +8mn3 + m4 + 4m2n2 +4m3n , Now we rearrange these terms , As :

Hypotenuse 2 = m4 + 2m3n + 2m2n2 + 2m3n + 4m2n2 +4 mn3 + 2m2n2 +4 mn3 + 4n4

Hypotenuse 2 = m2 ( m2 + 2mn + 2 n2 ) + 2mn ( m2 + 2mn + 2 n2 )+ 2n2 ( m2 + 2mn + 2 n2 )

Hypotenuse 2 = ( m2 + 2mn + 2 n2 ) ( m2 + 2mn + 2 n2 )

Hypotenuse 2 = ( m2 + 2mn + 2 n2 )2

Hypotenuse = ( m2 + 2mn + 2 n2 )

And

We know

Cosec θ = HypotenuseOpposite , Substitute both values , we get

Cosec θ = m2 + 2mn + 2n2m2 + 2mn ( Hence proved )