SOLUTION: If tan x + cot x = 2, then find the value tan^17 x + cot^17 x

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Question 1081080: If tan x + cot x = 2, then find the value tan^17 x + cot^17 x
Found 2 solutions by MathLover1, ikleyn:
Answer by MathLover1(20850) About Me  (Show Source):
You can put this solution on YOUR website!

tan%28x%29+%2B+cot%28x%29+=+2

sin%28x%29%2Fcos%28x%29+%2B+cos%28x%29%2Fsin%28x%29+=+2

%28sin%5E2+%28x%29+%2B+cos%5E2%28+x%29%29%2F%28sin%28x%29cos%28x%29%29+=+2

1%2F%28sin%28x%29cos%28x%29%29+=+2

2sin%28x%29cos%28x%29=+1

sin%28+2x+%29=+1

2x+=+90° for 0+%3C=+x+%3C=+180°

x+=+45° or pi%2F4 radians

we know tan+%2845%29+=+1 and cot%2845%29+=+1

then tan%5E17+%28x%29+%2B+cot%5E17%28+x%29=+1%5E17%2B+1%5E17=+2

Answer by ikleyn(52817) About Me  (Show Source):
You can put this solution on YOUR website!
.
If tan(x) + cot(x) = 2, then 

   sin%28x%29%2Fcos%28x%29 + cos%28x%29%2Fsin%28x%29 = 2,  ---->

   sin%5E2%28x%29%2F%28sin%28x%29%2Acos%28x%29%29 + cos%5E2%28x%29%2F%28sin%28x%29%2Acos%28x%29%29 = 2,  ---->

   %28sin%5E2%28x%29+%2B+cos%5E2%28x%29%29%2F%28sin%28x%29%2Acos%28x%29%29 = 2,  ---->

   1%2F%28sin%28x%29%2Acos%28x%29%29 = 2,

   2sin(x)*cos(x) = 1  --->  sin(2x) = 1  --->  2x = pi%2F2  --->  x = pi%2F4  or  x = 3pi%2F4. 

   In any case, tan(x) = 1  and  cot(x) = 1.


       So, we proved that  if  tan(x) + cot(x) = 2  then  tan(x) = 1  and  cot(x) = 1.



2.  Having this, you have  tan%5E17%28x%29+%2B+cot%5E17%28x%29 = 1%5E17+%2B+1%5E17 = 1 + 1 = 2.

Answer.  if tan(x) + cot(x) = 2  then  tan%5E17%28x%29+%2B+cot%5E17%28x%29 = 2.


Solved.