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Question

If tan(AB)=1 and sec(A+B)=23, then the smallest positive value of A and B, respectively, are

A
25π24,19π24
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B
19π24,25π24
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C
31π24,13π24
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D
13π24,31π24
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Solution

The correct option is A 25π24,19π24
tan(AB)=1
sec(A+B)=23tan(A+B)=13
tan2A=tan((A+B)+(AB))=tan(A+B)+tan(AB)1tan(A+B)tan(AB)
=13+1113=1+331=2+3
2A=nπ+5π12A=nπ2+5π24
tan2B=tan((A+B)(AB))=tan(A+B)tan(AB)1+tan(A+B)tan(AB)
=1311+13=133+1=23
2B=nπ+π12B=nπ2+π24
A=19π24,B=25π24

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