if `(tanalpha-i(sin ""(alpha)/(2)+cos ""(alpha)/(2)))/(1+2 i sin ""(alpha)/(2))` is purely imaginary then `alpha` is given by -

Question

if `(tanalpha-i(sin ""(alpha)/(2)+cos ""(alpha)/(2)))/(1+2 i sin ""(alpha)/(2))` is purely imaginary then `alpha` is given by -

if `(tanalpha-i(sin ""(alpha)/(2)+cos ""(alpha)/(2)))/(1+2 i sin ""(alpha)/(2))` is purely imaginary then `alpha` is given by -
A. `n pi +(pi)/(4)`
B. `n pi =(pi)/(4)`
C. `(2n+1) pi`
D. `2 n pi +(pi)/(4)`

Monjur Alahi

4 views

2025-01-04 04:42

1 Answers

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Salim Ahmed Kawsar · Accepted Answer · 2023-02-05T15:29:48+06:00

Correct Answer - A
`([tan alpha -i(sin ""(alpha)/(2)+cos""(alpha)/(2))])/((1+2i sin ""(alpha)/(2))).((1-2i sin ""(alpha)/(2)))/((1-2isin""(alpha)/(2)))`
`([tan alpha-2.sin ""(alpha)/(2)(sin""(alpha)/(2)+cos""(alpha)/(2))]+i(2 tanalpha(sin""(alpha)/(2))-(sin""(alpha)/(2)+cos""(alpha)/(2))))/((1+4sin^(2)""(alpha)/(2)))`
`implies tan alpha - 2 sin ""(alpha)/(2).(sin""(alpha)/(2)+cos""(alpha)/(2))=0`
` tan alpha = 2 sin ((alpha)/(2)).(sin""(alpha)/(2)+cos""(alpha)/(2))`
`(cos""(alpha)/(2))/(cosalpha)=(sin ""(alpha)/(2)+cos""(pi)/(2))`
By solving `alpha = n pi +(pi)/(4)`