I am trying to verify some trig identities but I am running into some problems.
If you could help me at all with any problem, I would really appreciate it, thanks!
It so happens that all 4 problems are easy and "short" to solve/verify, otherwise you should have posted them separtately. Asking for many problems in only one posting is too much if the problems are "hard" or "long" to solve.Originally Posted by lizzy20
If one doesn't have time to solve all of the problems, he might ignore all of the problems outright, and you'd get no answer from him. Or if it is me, even if I have enough time, if I cannot solve all of the problems, I will ignore all of them and I will bypass your question.
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[cscX] /[cot +tanX] = cosX
This kind, we convert all functions into their sine or cosine equivalents.
Lefthand Side, (LFS) =
= [cscX] /[cotX +tanX]
= [1 /sinX] /[(cosX/sinX) +(sinX/cosX)]
= [1 /sinX] /[(cosX*cosX +sinX*sinX) /(sinX*cosX)]
= [1 /sinX] /[(cos2(X) +sin^2(x)) /(sinXcosX)]
Since sin^2(X) +cos^2(x) = 1, then,
= [1 /sinX] /[1 /(sinXcosX)]
= [1 /sinX]*[(sinXcosX) /1]
= cosX
= Righthand Side, (RHS)
Verified.
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[1 /(sinY -1)] -[1 /(sinY +1)] = -2sec^2(Y)
LFS =
= [1 /(sinY -1)] -[1 /(sinY +1)]
Combine them into one fcraction only,
= [(sinY +1)*1 -(sinY -1)*1] /[(sinY -1)(sinY +1)]
= [sinY +1 -sinY +1] /[(sinY)^2 -1^2]
= [2] /[sin^2(Y) -1]
From sin^2(Y) +cos^2(Y) = 1, we can get sin^2(Y) = 1 -cos^2(Y), so,
= 2 /[(1 -cos^2(Y)) -1]
= 2 /[-cos^2(y)]
= -2sec^2(Y)
= RHS
Verified.
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sin^3(X) -cos^3(X) = (1 +sinXcosX)(sinX -cosX)
The LHS is a difference of cubes, in the form a^3 -b^3. Factoring that,
a^3 -b^3
= (a-b)(a^2 +ab +b^2)
So, adopting that to the LHS,
= (sinX -cosX)[sin^2(X) +sinXcosX +cos^2(X)]
= (sinX -cosX)[sin^2(X) +cos^2(X) +sinXcosX]
= (sinX -cosX)[1 +sinXcosX]
= (1 +sinXcosX)(sinX -cosX)
= RHS
Verified.
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[tan(theta)] +[(cos(theta)) /(1 +sin(theta))] = sec(theta)
Let X = theta, for less confusion, and easy typing.
[tanX] +[cosX /(1 +sinX)] = secX
LHS =
= tanX +[cosX /(1 +sinX)]
= [sinX /cosX] +[cosX /(1 +sinX)]
= [(1 +sinX)*sinX +cosX*cosX] /[cosX*(1 +sinX)]
= [sinX +sin^2(X) +cos^2(X)] /[cosX*(1 +sinX)]
= [sinX +1] /[cosX*(1 +sinX)]
The (sinX +1) or (1 +sinX) is called out,
= [1] /[cosX]
= secX
= RHS
Verified.
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I assume you know how to combine two fractions into one fraction only.
a/b +c/d
The common denominator is b*d,
= (d*a +b*c) /(b*d).
I am not trying to argue with ticbol he is correct. I just want to redo them in LaTeX form because I think it is easier to follow and I also need practice with my LaTeX code. I might do them out of order or not even all of them.
Problem 3 verify that:
$\displaystyle sin^3x-cos^3x=(1+sinxcosx)(sinx-cosx)$
by the difference of 2 cubes identity:
$\displaystyle a^3-b^3=(a-b)(a^2+ab+b^2)$
we have:
$\displaystyle sin^3x-cos^3x=(sinx-cosx)(sin^2x+sinxcosx+cos^2x)$
but: (pythagorean identities)
$\displaystyle sin^2x+cos^2x=1$
thus:
$\displaystyle sin^3x-cos^3x=(sinx-cosx)(1+sinxcosx)$
Verified
Problem 1
Verify that
$\displaystyle \frac{cscx}{cotx+tanx}=cosx$
First work with the denominator of the fraction:
$\displaystyle cotx+tanx=\frac{cosx}{sinx}+\frac{sinx}{cosx}$
Adding fraction the common denominator is:
$\displaystyle sinxcosx$
thus:
$\displaystyle \frac{sin^2x+cos^2x}{sinxcosx}$
but:
$\displaystyle sin^2x+cos^2x=1$
thus:
$\displaystyle \frac{1}{sinxcosx}$
is the simplified denominator.
Second the numerator of the fraction is:
$\displaystyle cscx=\frac{1}{sinx}$
Here is the most important step remember when you divide fractions you flip the one on the right and then multiply. Thus, the left hand side is a mess but it simplifies:
$\displaystyle \frac{\frac{1}{sinx}}{\frac{1}{sinxcosx}}$
Thus, you have:
$\displaystyle \frac{1}{sinx}\frac{sinxcosx}{1}$
Cancel $\displaystyle sinx$ and you are left with
$\displaystyle cosx$
Verified
Problem 2
Verified that
$\displaystyle \frac{1}{sinx-1}-\frac{1}{sinx+1}=-2sec^2x$
Remember you are adding fractions, the common denominator here is the product of the two fractions. Thus the common denominator is:
$\displaystyle (sinx-1)(sinx+1)=sin^2x+sinx-sinx-1=sin^2x-1$ by FOIL
But $\displaystyle 1-sin^2x=cos^2x$ By Pythagorean Identity thus:
$\displaystyle sin^2x-1=-cos^2x$
Now change to common denominator:
$\displaystyle \frac{sinx+1}{(sinx-1)(sinx+1)}-\frac{sinx-1}{(sinx-1)(sinx+1)}$
Bring them together because they have a common denominator:
$\displaystyle \frac{(sinx+1)-(sinx-1)}{(sinx-1)(sinx+1)}$
Open the parantheses ( watch the negative sign)
$\displaystyle \frac{sinx+1-sinx+1}{(sinx-1)(sinx+1)}$
But in the very beginning I explained that the common denominator
$\displaystyle (sinx-1)(sinx+1)$ is $\displaystyle -cos^2x$
But:
$\displaystyle \frac{1}{cosx}=secx$
Thus:
$\displaystyle \frac{2}{-cos^2x}=-2sec^2x$
Verified.
Problem 4)
Verify that
$\displaystyle tanx+\frac{cos}{1+sinx}=secx$
Express $\displaystyle tanx$ as
$\displaystyle \frac{sinx}{cosx}+\frac{cosx}{1+sinx}$
You are adding fractions again, thus find the common denominator which in this case this the product of the denominatos thus $\displaystyle cosx(1+sinx)$
Change fractions to common denominator
$\displaystyle \frac{sinx(1+sinx)}{cos(1+sinx)}+\frac{cosxcosx}{c osx(1+sinx)}$
Adding fraction because they have a common denominator and remembering that $\displaystyle cosxcosx=cos^2x$ you have:
$\displaystyle \frac{sinx(1+sinx)+cos^2x}{cosx(1+sinx)}$
Open parantheses:
$\displaystyle \frac{sinx+sin^2x+cos^2x}{cox(1+sinx)}$
But $\displaystyle cos^2x+sin^2x=1$
Thus:
$\displaystyle \frac{sinx+1}{cox(1+sinx)}$
Now cancel $\displaystyle 1+sinx$
$\displaystyle \frac{1}{cosx}$
But $\displaystyle secx=\frac{1}{cosx}$
Verified.