Given A + B + C = 180 <mrow class="MJX-TeXAtom-ORD"> &#x2218

Brunton39

Brunton39

Answered question

2022-06-21

Given A + B + C = 180 , find value of tan A tan B + tan B tan C + tan A tan C sec A sec B sec C

Answer & Explanation

drumette824ed

drumette824ed

Beginner2022-06-22Added 19 answers

If you write tan C = tan ( π ( A + B ) ) = tan ( A + B ), you have
tan A tan B + tan B tan C + tan C tan A = tan A tan B ( tan A + tan B ) tan A + tan B 1 tan A tan B = tan A tan B tan 2 A tan 2 B tan 2 A 2 tan A tan B tan 2 B 1 tan A tan B = ( tan 2 A tan 2 B + tan 2 A + tan 2 B + 1 ) + ( tan A tan B 1 ) tan A tan B 1 = 1 tan A tan B 1 ( tan 2 A + 1 ) ( tan 2 B + 1 ) + 1 = cos A cos B sin A sin B cos A cos B 1 cos 2 A 1 cos 2 B + 1 = 1 cos A cos B cos C + 1
because sin A sin B cos A cos B = cos ( A + B ) = cos C

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