If \frac{1-\sin x}{1+\sin x}=4, what is the value of \tan

Monique Slaughter

Monique Slaughter

Answered question

2021-12-31

If 1sinx1+sinx=4, what is the value of tanx2?
1)-3
2)2
3)-2
4)3

Answer & Explanation

Bertha Jordan

Bertha Jordan

Beginner2022-01-01Added 37 answers

Let t be t=tan(x2) for the "good x" satisfying the given relation. Then sinx=2t1+t2, so
4=1sinx1+sinx=(1+t2)2t(1+t2)+2t=(1t1+t)2
This gives for 1t1+t the values ±2, leading to the two solutions -3 and -1/3
maul124uk

maul124uk

Beginner2022-01-02Added 35 answers

You have
1sinx1+sinx=4sin2(x2)+cos2(x2)2sin(x2)cos(x2)sin2(x2)+cos2(x2)+2sin(x2)cos(x2)=4
(sin(x2)cos(x2)sin(x2)+cos(x2))2=4
sin(x2)cos(x2)sin(x2)+cos(x2)=±2
tan(x2)1tan(x2)+1=±2
The only solution of the equation tan(x2)1tan(x2)+1=2 is tan(x2)=3 which is on that list, whereas the only solution of the equation tan(x2)1tan(x2)+1=2 is tan(x2)=13, which is not on that list.
So, the problem has two solutions, but only one of them is on the list of options.
karton

karton

Expert2022-01-08Added 613 answers

tanx2=sinx2cosx2=2sinx2cosx22cos2x2=sinxcosx+1
hence with sinx=35, cosx=±1sin2x=±45, tanx2=13, 3

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