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Question

(i) Show that tan9otan27otan63o+tan81o=4
(ii) Show that cos42o+cos78o+cos162o=0

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Solution

(i) We know that tanθ=cot(90θ)
=> tan81=cot9
and tan63=cot27

We need to simplify

tan9+cot9tan27cot27
=> sin9cos9+cos9sin9sin27cos27cos27sin27
=>1sin9.cos91sin27.cos27
=>22sin9.cos922sin27.cos27
=>2sin182sin54

Putting values of sin18=514,sin54=5+14 and simplying, we get the expression equal to 4.

(ii) cos42+cos78+cos162
=> cos(6018)+cos(60+18)+cos162
=> 2cos60.cos18+cos(18018)
=>212cos18cos18
=> 0

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