Verify trig equation

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• Nov 10th 2012, 06:50 PM
Darion
Verify trig equation

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

1 + sec2x sin2x = sec2x
• Nov 10th 2012, 07:17 PM
MarkFL
Re: Verify trig equation
I assume you mean:

$1+\sec^2(x)\sin^2(x)=\sec^2(x)$

Right?
• Nov 10th 2012, 07:24 PM
Darion
Re: Verify trig equation
Yes. Sorry.
• Nov 10th 2012, 07:38 PM
MarkFL
Re: Verify trig equation
Hey, no problem, just wanted to make sure, and just use the caret symbol "^" to denote exponentiation.

You should be familiar with a Pythagorean identity involving the secant function. That along with rewriting $\sec^2(x)\sin^2(x)$ (can you think of a simpler way to write this term?) should help quite a bit.
• Nov 10th 2012, 07:51 PM
Darion
Re: Verify trig equation
The only Pythagorean identity in my book (that I can find at least) is sin^2u+cos^2u=1
• Nov 10th 2012, 07:57 PM
MarkFL
Re: Verify trig equation
You can derive two other useful forms of this identity by dividing through by either $\sin^2(u)$ or $\cos^2(u)$.
• Nov 10th 2012, 08:26 PM
Darion
Re: Verify trig equation
So then I divide both sides by sin^2x?
• Nov 10th 2012, 08:35 PM
MarkFL
Re: Verify trig equation
You want the form involving the square of the secant function, and since the right side of the Pythagorean identity you cited is 1, you should divide through by $\cos^2(u)$, or equivalently, multiply through by $\sec^2(u)$. In fact, if you do the latter, you get the very equation you are given to verify!
• Nov 10th 2012, 08:38 PM
Darion
Re: Verify trig equation
OK I'm kind of lost now. I see that (and why) we want to use cos^2x, but I'm not sure how to divide that through.
• Nov 10th 2012, 09:32 PM
MarkFL
Re: Verify trig equation
I would begin with the Pythagorean identity:

$\cos^2(x)+\sin^2(x)=1$

Now, multiply through by $\sec^2(x)$, that is, multiply every term on both sides by this. What do you get. Recall:

$\sec(x)\equiv\frac{1}{\cos(x)}$
• Nov 10th 2012, 09:52 PM
Darion
Re: Verify trig equation
OK so we can turn it into

1 + (1/cos^2x) (sin^2x)^2 = (1/cos^2x)

I'm hopeful :P
• Nov 10th 2012, 09:56 PM
MarkFL
Re: Verify trig equation
I only wanted you to use the definition of the secant function for the first term on the left to get 1, as you did, but for the other terms, just use the square of the secant function, and you will find you will get the equation you are trying to verify.
• Nov 10th 2012, 10:00 PM
Darion
Re: Verify trig equation
Oh ok so its just 1 + (1/cos^2x) (sin^2x)^2 = sec^2x
• Nov 10th 2012, 10:05 PM
MarkFL
Re: Verify trig equation
Don't write (1/cos^2(x)) on the left side, write sec^2(x), and you then have the exact equation you are trying to verify.
• Nov 10th 2012, 10:13 PM
Darion
Re: Verify trig equation
So we are just back at 1 + sec^2x sin2x = sec^2x
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