value of 2 \tan^{-1}(\csc \alpha)+\tan^{-1}(2\sin \alpha \sec^2 \alpha)

Seamus Kent

Seamus Kent

Answered question

2022-01-24

value of 2tan1(cscα)+tan1(2sinαsec2α)

Answer & Explanation

Dakota Cunningham

Dakota Cunningham

Beginner2022-01-25Added 9 answers

Note that
2sinαsec2α=2sinαcos2α=2sinα1sin2α
If
f(x)=arctan2x1x2
then
f(x)=21+x2
so f(x) differs from 2arctanx by a constant over (−1,1). Since f(0)=0=2arctan0, we can say that
arctan2sinα1sin2α=2arctansinα
For x>0, arctan(1x)=π2arctanx, so your expression evaluates to
2(π2arctansinα)+2arctansinα=π
if sinα>0
If sinα<0 , the expression evaluates to π, because for x<0 one has arctan(1x)=π2arctanx

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