A-b-a-b-tanA-B-2-tanA-B-2

A b/a+b=tan(A B/2 )/tan(A+B/2 ) Let a = k sin A, b = k sin B (Using sine rule) LHS =a b/a+b=K sin A K sin B/K sin A + K sin B =sin A ...

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Standard XII

Mathematics

Trigonometric Ratios Using Right Angled Triangle

a-b/a+b=tanA-...

Question

$\frac{a - b}{a + b} = \frac{t a n (\frac{A - B}{2})}{t a n (\frac{A + B}{2})}$

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Solution

$\frac{a - b}{a + b} = \frac{t a n (\frac{A - B}{2})}{t a n (\frac{A + B}{2})} Let a = k sin A, b = k sin B (Using sine rule) LHS = \frac{a - b}{a + b} = \frac{K sin A - K sin B}{K s i n A + K s i n B} = \frac{sin A - sin B}{sin A + sin B} = \frac{2 c o s (\frac{A + B}{2}) s i n (\frac{A - B}{2})}{2 s i n (\frac{A + B}{2}) c o s (\frac{A - B}{2})} = \frac{t a n (\frac{A - B}{2})}{t a n (\frac{A + B}{2})} = R H S$

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a−ba+b=tan(A−B2)tan(A+B2)

a−ba+b=tan(A−B2)tan(A+B2)Let a = k sin A, b = k sin B (Using sine rule)LHS=a−ba+b=K sin A - K sin BKsinA+KsinB=sin A - sin Bsin A + sin B=2cos(A+B2)sin(A−B2)2sin(A+B2)cos(A−B2)=tan(A−B2)tan(A+B2)=RHS

$\frac{a - b}{a + b} = \frac{t a n (\frac{A - B}{2})}{t a n (\frac{A + B}{2})}$