If (cosec∅-sin∅)=a^{3} and (sec∅-cos∅)=b^{3} , prove that a^{2} b^{2} (a^{2} +a^{2} )=1. Created by takshnaria58.

If (cosec∅-sin∅)=a^{3} and (sec∅-cos∅)=b^{3} , prove that a^{2} b^{2} (a^{2} +a^{2} )=1

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September 2023 1 2 Report

If (cosec∅-sin∅)=a^{3} and (sec∅-cos∅)=b^{3} , prove that a^{2} b^{2} (a^{2} +a^{2} )=1

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SHARDULfrefirelegend Given : cosecθ−sinθ=a
3
and secθ−cosθ=b
3

⇒
sinθ
1

−sinθ=a
3

⇒
sinθ
cos
2
θ

=a
3

⇒a=(
sinθ
cos
2
θ

)
3
1

Similarly ; b=(
cosθ
sin
2
θ

)
3
1

⇒a
2
b
2
(a
2
+b
2
)=a
4
b
2
+a
2
b
4

⇒(
sinθ
cos
2
θ

)
3
4

(
cosθ
sin
2
θ

)
3
2

+(
sinθ
cos
2
θ

)
3
2

(
cosθ
sin
2
θ

)
3
4

⇒(cosθ)
8/3−2/3
.(sinθ)
4/3−4/3
+(sinθ)
8/3−2/3
.(cosθ)
4/3−4/3

⇒cos
2
θ+sin
2
θ=1

⇒a
2
b
2
(a
2
+b
2
)=

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