If `sin alpha=A sin(alpha+beta),Ane0`, then
Which of the following is not the value of `tan(alpha+beta)`?

Options

A. `(sin beta)/(cos beta-A)`
B. `(sin alpha cos alpha)/(A cos beta-sin^(2)alpha)`
C. `(sin alpha cos alpha)/(A cos beta+sin^(2)alpha)`
D. none of these.

Answer

C

Explanation

`tan(alpha + beta) = (tan alpha +tan beta)/(1-tan alpha tan beta)`
`" " = ((A sin beta)/(1-A cosebeta)+ ( sin beta)/( cos beta))/(1-(A sin betasinbeta)/((1-Acosbeta)cosbeta))`
`" " = (Asin beta cos beta + sin beta -A sin beta cos beta)/(cos beta -Acos^(2) beta -A sin^(2) beta)`
`" " = (sin beta)/(cos beta -A)`
Also, `tan(alpha +beta)= (tan alpha + tan beta)/( 1-tan alpha tan beta)`
`" " = ((sin alpha)/(cosalpha) + (sinalpha (1-A cos beta))/(A cos alpha cos beta))/(1-(sin ^(2)alpha (1-A cos beta))/(A cos ^(2) alpha cos beta))` [from Eq. (ii)]
`" " = [ A sin alpha cos beta + sin alpha - A sin alpha cos beta]cos alpha)/(A cos^(2) alpha cos beta - sin ^(2) alpha + A sin^(2) alpha cos beta)`
`" " = ( sin alpha cos alpha )/( A cos beta - sin ^(2)alpha )`