Question

Can someone please help me?

Verify each trigonometric equation by substituting identities to match the right hand side of the equation to the left hand side of the equation.

cot x sec4x = cot x + 2 tan x + tan3x

(sin x)(tan x cos x - cot x cos x) = 1 - 2 cos2x

1 + sec2x sin2x = sec2x

[(sin(x))/(1-cos(x))]+[(sin(x))/(1+cos(x))]=2csc(x)

- tan2x + sec2x = 1

Answer 1

@Darion whenever you're ready!

Answer 2

I'm ready. :P

Answer 3

Is that first one\[\cot x \sec 4x =\cot x +2\tan x +\tan 3x\]or\[\cot x \sec ^{4}x =\cot x +2\tan x +\tan ^{3}x\]@Darion

Answer 4

The second one you posted. Thanks for replying man

Answer 5

No problem and yes my teacher instincts told me that it's the second one but I just wanted to check. OK so first step is to choose a side to begin with. Choose the side which you think can easily be manipulated. Which side waould you say we should begin with?

Answer 6

@Darion I am a teacher in real life. LOL Therefore I'm not just going top give you the answers. if you want my help then please reply to my questions. If you don't know the answer then don't be embarrassed to say so but don't keep me waiting because I don't have time for that. Understood?!

Answer 7

The right side would be easier to manipulate

Answer 8

Now let's try this one more time, We have\[\cot x \sec ^{4}x =\cot x +2\tan x +\tan ^{3}x\]We'll start with the right side. Good! Now it is usually wise when proving identities to first rewrite every term in terms of sine and cosine. Hence first rewrite the right side in terms of sine and cosine. Go ahead.

Answer 9

That is really where my main problem is at; rewriting all these different identities

Answer 10

I don't know how to do that

Answer 11

OK here are the basic identities you should know

PYTHAGOREAN IDENTITIES\[\sin ^{2}\theta +\cos ^{2}\theta =1\]
\[1+\tan ^{2}\theta =\sec ^{2}\theta\]
\[\cot ^{2}\theta +1=\csc ^{2}\theta\]
QUOTIENT IDENTITIES\[\tan \theta =\frac{ \sin \theta }{ \cos \theta }\]
\[\cot \theta =\frac{ \cos \theta }{ \sin \theta }\]
RECIPROCAL IDENTITIES\[\csc \theta =\frac{ 1 }{ \sin \theta }\]
\[\sec \theta =\frac{ 1 }{ \cos \theta }\]
\[\cot \theta =\frac{ 1 }{ \tan \theta }\]There are many more but base on your questions, I think these are all you need to know for now. Have you learned these?

Answer 12

I'll do the the first one as an example, then you can do the others. Prove\[\cot x \sec ^{4}x =\cot x +2\tan x +\tan ^{3}x\]
Proof:\[R.S.=\cot x +2\tan x +\tan ^{3}x\]
\[=\frac{ \cos x }{ \sin x }+\frac{ 2\sin x }{ \cos x }+\frac{ \sin ^{3}x }{ \cos ^{3}x }\]
\[=\frac{ \cos ^{4}x +2\sin ^{2}x \cos ^{2}x +\sin ^{4}x }{ \sin x \cos ^{3}x }\]
\[=\frac{ (\cos ^{2}x +\sin ^{2}x)^{2} }{ \sin x \cos ^{3}x }\]
\[=\frac{ 1 }{ \sin x \cos ^{3}x }\]
\[=\frac{ \cos x }{ \sin x \cos ^{4}x }\]
\[=\frac{ \cos x }{ \sin x }⋅\frac{1 }{ \cos ^{4}x }\]
\[=\cot x \sec ^{4}x\]
\[=L.S.\]∴ L.S. = R.S.
∴ QED

@Darion I hope you understand and try the others.

Answer 13

Thanks for the list of identities. That helps a lot. I think I understand this...sort of. Give me a minute to try the next one.

Answer 14

how did you get cosx from 1

Answer 15

Good question. I multiplied both the numerator and the denominator by cos x, thus unchanging the expression in the previous step. Thanks for asking.

Answer 16

verify csc squared theta/cot theta=csc theta sec theta