Question

tanA+2tan2A+4cot4A=prove that cotA​

Answer 1

tan 2A = (2tanA) / (1- tan²A)

So, (2) tan 4A = ( 2tan2A) / (1- tan² 2A)

LHS = tanA + 2tan2A + 4/(tan4A)

= tanA + 2tan2A + {4/ (2tan2A)/(1-tan²2A)}….by using (2)nd function

= tanA + 2tan2A + 4(1-tan²2A) / 2tan2A

= tanA + 2tan2A + 2(1-tan²2A) / tan2A

= tanA + {( 2tan² 2A + 2 - 2tan²2A)} / tan2A

= tanA + { ( 2/ tan2A ) }

= tanA + [2 / {2tanA/(1-tan² A)}]…. by using 1st function

= tanA + [ {2( 1-tan²A)} /2tanA ]

= {(2tan² A + 2 - 2tan² A )} / 2tanA

= 2/ (2 tanA)

= 1/tanA

= cot A = RHS

[ Hence Proved]