Question 885010: if tan theta+cot theta=2,find the value of tan7theta+cot7theta.(Sir please explain how to solve problems which have number between sin,cos,tan and theta like tan7theta)
Answer by KMST(5345)   (Show Source):
You can put this solution on YOUR website! THE ANSWER:
REQUESTED EXPLANATION (not a direct way to solve this problem):
When I have to deal with trigonometric functions of sums or multiples of angles,
I look for a table of trigonometric identities.
Trigonometric identities are relations (formulas) like
 .
There are many more of those.
(I do not have those formulas memorized, and
although I can rediscover trig identity formulas,
it is much faster to look them up).
Your teacher may believe otherwise, but as professor Peter Alfeld says in his U. of Utah website,
"Understanding mathematics requires no memorization whatsoever".
(I never met him or stepped into U. of Utah,
but I found his website when looking for someone who agreed with my philosophy.
I do not like to memorize what I can rediscover/re-deduce/calculate by myself.
What I cannot calculate, like phone numbers,
if possible, I write down and/or look up.
I prefer to remember facts and formulas without trying,
just letting the stuff get ingrained in my memory from repeated use).
THE SOLUTION:
You are unlikely to find trig identities dealing with  is a very special angle, that should be very familiar to you.
You are given 
 ,
 ,
and  .
Those formulas appear in listings of trig identities,
but are used so often that I do not need to look them up.
You can use
 to find
 .
That is easy enough, because with  the equation turns into
 ---> ---> ---> ---> ---> ---> .
From there you realize that
 for any integer 
(or  if you like degrees).
You can also work with the sine and cosine trigonometric identity formulas to find  :
 .
So  ---> 
Now, we use the trig identity :
--->  .
The angle between  and  whose sine is  is 
(or  if you like degrees better than radians),
but all angles coterminal with have a sine of , so
 for any integer .
So for any integer
(or if you like degrees).
Any way you get there,
for any integer
(or if you like degrees).
Since tangent and cotangent have a period of  or  ,
tangent and cotangent have the same value for all those angles.
They also have the same value for all angles represented by
 ,
so we can work with  (or  if you like).
You may know that
 and  ,
and since  .
You can use  and 
to calculate that 
You can also use
to justify that,
since is coterminal with  ,
implies .
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