Question

if X = a sin theta and y = b tan theta
then prove that
a^2 / x^2 - b^2 / y^2 = 1​

Answer 1

Step-by-step explanation:

We have , 

L.H.S=x2a2−y2b2

⇒L.H.S=a2sin2θa2−b2tan2θb2     [∵x=asinθ,y=btanθ]

⇒L.H.S=sin2θ1−tan2θ1

⇒L.H.S=cosec2θ−cot2θ     [∵1+cot2θ=cosec2θ∴cosec2θ−cot2θ=1]

⇒ LHS =1= RHS

Answer 2

Answer:

= 1 ( Proved ).

Step-by-step explanation:

X = a sin theta

Y= b tan theta

Prove: a² / x² - b² / y² = 1.

= a² / ( a sin theta )² - b² ( b tan theta )²

= a² / ( a² sin² theta ) - b² / ( b² tan² theta )

= a² / ( a² sin² theta ) - b² / b² ( sin² theta / cos² theta ) ( As tan² theta = sin² theta / cos² theta ) ( Cut a² with a² and b² with b² )

= 1 / sin² theta - ( 1 / sin² theta ) / cos² theta

= 1 / sin² theta - ( 1 × cos² theta ) / sin² theta

= 1 / sin² theta - cos² theta / sin² theta

= ( 1 - cos² theta ) / sin² theta

= sin² theta / sin² theta ( As sin² theta = ( 1 - cos² theta ) ( Cut sin² theta with sin² theta )

= 1 ( Proved ).

Thus, ( L.H.S. = R.H.S. ).

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