Tangent, Cotangent, Secant, Cosecant in Terms of Sine and Cosine
If tan A=x si...
Question
If tanA=xsinB1−xcosB and tanB=ysinA1−ycosA, then the value of sinAsinB for all permissible values of A,B is
A
xy
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B
x+yx−y
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C
yx
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D
y+xy−x
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Solution
The correct option is Axy tanA=xsinB1−xcosB⇒sinAcosA=xsinB1−xcosB⇒sinA−xcosBsinA=xsinBcosA⇒sinA=x(cosAsinB+cosBsinA)⇒sinA=xsin(A+B)⋯(1) tanB=ysinA1−ycosA⇒sinBcosB=ysinA1−ycosA⇒sinB−ycosAsinB=ycosBsinB⇒sinB=y(cosAsinB+cosBsinB)⇒sinB=ysin(A+B)⋯(2) Now, sinAsinB=xsin(A+B)ysin(A+B)∴sinAsinB=xy