Question
If cos 2 B =
cos(A+C)cos(A−C), then tan A, tan B, tan C are in
cos(A+C)cos(A−C), then tan A, tan B, tan C are in
Answer: Option B
:
B
cos 2B =
cos(A+C)cos(A−C) =
cosAcosC−sinAsinCcosAcosC+sinAsinC
⇒1−tan2B1+tan2B =
1−tanAtanC1+tanAtanC
⇒1+tan2B−tanAtanC−tanAtanCtan2B
= 1−tan2B+tanAtanC−tanAtanCtan2B
⇒ 2tan2B=2tanAtanC⇒ tan2B = tan A tan C
Hence, tan A, tan B and tan C will be in G.P.
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:
B
cos 2B =
cos(A+C)cos(A−C) =
cosAcosC−sinAsinCcosAcosC+sinAsinC
⇒1−tan2B1+tan2B =
1−tanAtanC1+tanAtanC
⇒1+tan2B−tanAtanC−tanAtanCtan2B
= 1−tan2B+tanAtanC−tanAtanCtan2B
⇒ 2tan2B=2tanAtanC⇒ tan2B = tan A tan C
Hence, tan A, tan B and tan C will be in G.P.
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