1.

If `(1+sint)(1+cost)=5/4`then find the value of `(1-sint)(1-cost)dot`

Answer» `(1+sint)(1+cost) = 5/4`
`=>1+sint+cost+sintcost = 5/4`
`=>(sint+cost) = 1/4-sintcost->(1)`
`=>(sint+cost)^2 = (1/4-sintcost)^2`
`=>sin^2t+cos^2t+2sintcost = 1/16+(sintcost)^2 -(sintcost)/2`
`=>1+sin2t = 1/16+(sin^2 2t)/4 - (sin2t)/4`
`=>16+16sin2t = 1+4sin^2 2t -4sin2t`
`=>4sin^2 2t -20sin2t -15 = 0`
`:. sin 2t = (20+-sqrt(400-4(-15)(4)))/8 = (5-2sqrt10)/2`
Here, we have rejected the value `(5+2sqrt10)/2` as it becomes more than `1`.
Now, `(1-sint)(1-cost) = 1-(sint+cost)+sintcost`
`=1-(1/4-sintcost)+sintcost`
`=3/4+sin2t`
`=3/4+5/2-sqrt10`
`=13/4-sqrt10`
`:. (1-sint)(1-cost) = 13/4 - sqrt10.`


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