Question

How do i solve these trig identities?
(1 + cosx)/sinx = cot(x/2)

cos^6x + sin^6x = cos2x(1-(1/4)sin^2 2x)

also, how can I work with Eg. Tan4x, sin2^2x etc?

Answer 1

also, how can I work with Eg. Tan4x, sin2^2x etc?
Use double angle formulas and sin(2x) = 2sin(x)cos(x)

Answer 2

\[\cot (x/2) = 1/\tan (x/2)=1+\cos(x)/\sin(x)\]

Answer 3

Is the right side of that second identity \[\cos(2x)(1-\frac14\sin ^{2}(2x))\] ?

Answer 4

thank you

Answer 5

Could you please answer my question?

Answer 6

Yes it is

Answer 7

Are you sure it is cos^6x + sin^6x? Because I get cos^6x - sin^6x

Answer 8

Oops my bad, it's cos6x - sin^6x

Answer 9

How do you deal with the 1/4sin^2x?

Answer 10

I will show you.

Answer 11

\[\cos (2x)[1-\frac14\sin ^{2}(2x)]\]

Answer 12

\[(\cos ^{2}x - \sin ^{2}x)[1/\frac14(2\sin x \cos x)^{2}\]

Answer 13

That should be 1-1/4 of course

Answer 14

\[(\cos ^{2}x - \sin ^{2}x)[1- \frac14(4\sin ^{2}x \cos ^{2}x)]\]

Answer 15

\[(\cos ^{2}x - \sin ^{2}x)[1-\sin ^{2}x \cos ^{2}x]\]

Answer 16

\[(\cos ^{2}x - \sin ^{2}x)[1-\sin ^{2}x(1-\sin ^{2}x)]\]

Answer 17

\[(\cos ^{2}x - \sin ^{2}x)[1-\sin ^{2}x +\sin ^{4}x]\]

Answer 18

Now replace 1 with sin^2x + cos^2x

Answer 19

\[(\cos ^{2}x - \sin ^{2}x)[\sin ^{2}x + \cos ^{2}x - \sin ^{2}x + \sin ^{4}x]\]

Answer 20

\[(\cos ^{2}x - \sin ^{2}x)[\cos ^{2}x + \sin ^{4}x\]

Answer 21

now use FOIL and multiply the two binomials

Answer 22

\[\cos ^{4}x + \cos ^{2}x \sin ^{4}x - \sin ^{2}x \cos ^{2}x - \sin ^{6}x\]

Answer 23

Group the middle two terms and factor out cos^2xsin^2x

Answer 24

\[\cos ^{4}x + \cos ^{2}x \sin ^{2}x(\sin ^{2}x - 1)\]

Answer 25

Now replace the sin^2x inside the parentheses with 1 - cos^2x

Answer 26

\[\cos ^{4}x+\cos ^{2}x \sin ^{2}x(1-\cos ^{2}x -1)\]

Answer 27

\[\cos ^{4}x +\cos ^{2}x \sin ^{2}x(-\cos ^{2}x)\]

Answer 28

\[\cos ^{4}x-\cos ^{4}x \sin ^{2}x - \sin ^{6}\]x

Answer 29

\[\cos ^{4}x(1-\sin ^{2}x)- \sin ^{6}\]

Answer 30

\[\cos ^{4}x(\cos ^{2}x)- \sin ^{6}x\]

Answer 31

You probably figured out that I dropped the sin^6 for a few steps. You should drag it along.

Answer 32

Thank you very much!

Answer 33

Do you understand?

Answer 34

Yes

Answer 35

That was a hard one!!!