If = sin cos Φ and = sin sin Φ then prove that
J (x,y/,Φ)×J(,Φ/x,y)=1
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Step-by-step explanation:
sinθ+sinϕ=a ________ (1) cosθ+cosϕ=b _______ (2)
(1)
2
+(2)
2
⇒a
2
+b
2
=sin
2
θ+sin
2
ϕ+2sinθsinϕ+cos
2
θ+cos
2
ϕ+2cosθcosϕ
(a
2
+b
2
)=1+1+2[sinθsinϕ+cosθcosϕ]
(a
2
+b
2
)=2[1+cos(θ−ϕ)]
2
a
2
+b
2
=2cos
2
(
2
θ−ϕ
)
⇒cos
2
(
2
θ−ϕ
)=
4
a
2
+b
2
______ (2)
tan(
2
θ−ϕ
)=
cos
2
(
2
θ−ϕ
)
1−cos
2
(
2
θ−ϕ
)
=
(
4
a
2
+b
2
)
1−(
4
a
2
+b
2
)
=
a
2
+b
2
4−a
2
−b
2
∴tan(
2
θ−ϕ
)=
a
2
+b
2
4−a 2 −b 2
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