1.

If `sin(alpha + beta) = 1, sin(alpha - beta) = 1/2` then `tan(alpha + 2 beta) tan(2 alpha + beta)`=

Answer» `sin(alpha+beta) = 1, sin(alpha-beta) = 1/2`
`alpha+beta = pi/2->(1)`,
` alpha-beta = pi/6->(2)`
Adding (1) and (2),
`alpha+beta+alpha-beta = pi/2+pi/6`
`=>2alpha = (4pi)/6`
`=>alpha = pi/3`
Putting value of `alpha` in (1),
`pi/3+beta = pi/2`
`=>beta = pi/2-pi/3 =>beta = pi/6`
`:. tan(alpha+2beta)tan(2alpha+beta) = tan(pi/3+pi/3)tan((2pi)/3+pi/6)`
`=tan((2pi)/3)tan((5pi)/6)`
`=(-sqrt3)(-1/sqrt3) = 1`


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