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Question

If cos(θα)=a,sin(θβ)=b, where (θα,θβ)(0,π2) then-

A
sin(αβ)=ab(1a2b2+a2b2)
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B
cos(αβ)=a(1b2)+b(1a2)
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C
sin(αβ)=ab+(1a2b2+a2b2)
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D
cos(αβ)=a(1b2)b(1a2)
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Solution

The correct options are
A sin(αβ)=ab(1a2b2+a2b2)
B cos(αβ)=a(1b2)+b(1a2)
Given cos(θα)=a,sin(θβ)=b, where (θα,θβ)ϵ(0,π2)
Now, sin(αβ)=sin((θβ)(θα))
=sin(θβ)cos(θα)cos(θβ)sin(θα)
=ab1a21b2
sin(αβ)=ab(1a2b2+a2b2)
cos(αβ)=cos((θβ)(θα))
=cos(θβ)cos(θα)+sin(θβ)sin(θα)
=a1b2+b1a2
cos(αβ)=a(1b2)+b(1a2)

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