Trigonometry> If eight theta is equal to pi show that c...

If eight theta is equal to pi show that cos 7 theta+ cot theta is equal to zero

akanksha , 7 Years ago

Grade 12th pass

3 Answers

Vikas TU

Last Activity: 7 Years ago

Dear Student,

The question should be cos 7 theta+cos theta=0.

8*theta=pi, so theta=pi/8.
Now, from half angle formula cos(theta/2) =± sqrt((1+ cos theta)/2).
Here pi/8 = ½ (pi/4) so we can apply half angle formula and
get cos(pi/8) = +sqrt((1+cos pi/4)/2)
= sqrt((1+(1/sqrt 2))/2
)=sqrt((2+sqrt2)/2) –(1)

Similarly for 7pi/8, applying half angle formula.
Here 7pi/8 = ½ (7pi/4) and
cos(7pi/4)=cos(2pi-pi/4)
=cos(pi/4
)=1/sqrt(2)

So cos(7pi/8) = -sqrt((1+cos 7pi/4)/2)[negative as 7pi/8 lies in second quadrant where cos is negative] =

-sqrt((1+(1/sqrt 2))/2)=
-sqrt((2+sqrt2)/2)—(2)

therefore , from (1) and (2) we get cos 7pi/4 + cos pi/4 =0

or cos 7theta + cos theta=0 (proved)

Cheers!!

Regards,

Vikas (B. Tech. 4^th year
Thapar University)

Aditya Kumar

Last Activity: 7 Years ago

Q. Should be Cos 7 theta+ cos theta=Cos (8 theta- theta) + cos theta=(Cos 8 theta × cos theta - sin 8 theta × sin theta) + cos theta=(Cos π × cos theta- sin π ×sin theta) +cos theta=(-1.cos theta- 0.sin theta) + cos theta=-cos theta +cos theta=0