Help with exact value of: \(\displaystyle{\tan{{\left({{\sin}^{{-{1}}}{\left(-\frac{{1}}{{2}}\right)}}-{{\tan}^{{-{1}}}{\left(\frac{{3}}{{4}}\right)}}\right)}}}\)

Petrolovujhm

Petrolovujhm

Answered question

2022-03-27

Help with exact value of:
tan(sin1(12)tan1(34))

Answer & Explanation

Drake Huang

Drake Huang

Beginner2022-03-28Added 15 answers

Let a=sin1(12) and b=tan1(34). We have
sin2a+cos2a=114+cos2a=1cosa=32
tana=33
since sin1x has [π2,π2] as image.
Also,
sinbcosb=34916cos2b+cos2b=1cosb=45sinb=35
since tan1x has (π2,π2) as image.
we then have
tan(ab)=tanatanb1+tanatanb=333413334=43+91233
=43+9123312+3312+33
=108+483+273+3614427
=144+753117
=48+25339

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