Assertion -In a triangle ABC if tanA-2-5-6-tanB-2-20-37- then tanC-2-2-5- Reason- tanC-2- A-B-2

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Sin(A+B)Sin(A-B)

Assertion :In...

Question

Assertion :In a triangle ABC if $tan \frac{A}{2} = \frac{5}{6}, tan \frac{B}{2} = \frac{20}{37}$ , then $tan \frac{C}{2} = \frac{2}{5}$ . Reason: $tan \frac{C}{2} = cot (\frac{A + B}{2})$

Both Assertion and Reason are correct and Reason is the correct explanation for Assertion

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Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion

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Assertion is correct but Reason is incorrect

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Assertion false but Reason is true

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Solution

The correct option is A Both Assertion and Reason are correct and Reason is the correct explanation for Assertion
In $△ A B C$
$A + B + C = π \Rightarrow \frac{A + B + C}{2} = \frac{π}{2} \frac{C}{2} = π - \frac{A + B}{2} tan \frac{C}{2} = tan (π - \frac{A + B}{2}) = c o t (\frac{A + B}{2}) = \frac{1 - tan \frac{A}{2} tan \frac{B}{2}}{tan \frac{A}{2} + tan \frac{B}{2}} = \frac{1 - \frac{5}{6} . \frac{20}{37}}{\frac{5}{6} . \frac{20}{37}} = \frac{2}{5}$

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Assertion :In a triangle ABC if tanA2=56,tanB2=2037, then tanC2=25. Reason: tanC2=cot(A+B2)

The correct option is A Both Assertion and Reason are correct and Reason is the correct explanation for AssertionIn △ABC A+B+C=π⇒A+B+C2=π2C2=π−A+B2tanC2=tan(π−A+B2)=cot(A+B2)=1−tanA2tanB2tanA2+tanB2=1−56.203756.2037=25

Assertion :In a triangle ABC if $tan \frac{A}{2} = \frac{5}{6}, tan \frac{B}{2} = \frac{20}{37}$ , then $tan \frac{C}{2} = \frac{2}{5}$ . Reason: $tan \frac{C}{2} = cot (\frac{A + B}{2})$