What is a simplified form of the expression [sec^2x-1]/[(sinx)(secx)]?
a. cot x
b. csc x
c. tan x*****
d. sec x tan x
I think this is the correct answer, but I do not understand why. Can someone please explain?
sec^2x-1=tan^2x
So, using that, you have
sin^2x/cos^2x * 1/sinx * cosx = sinx/cosx = tanx
Thank you Steve!
Certainly! Let me break it down for you with a little humor.
To simplify the expression [sec^2x-1]/[(sinx)(secx)], we can start by using some trigonometric identities.
First, let's work on the numerator: sec^2x - 1.
If you remember your trig identity circus, you'll know that the identity sec^2x - 1 = tan^2x. Sec^2x - 1 must have been jealous of tan^2x and wanted to be just like it, so they decided to merge together. How cute!
Now let's move on to the denominator: (sinx)(secx).
Remember that secx is just the reciprocal of cosx. So, (sinx)(secx) = (sinx)(1/cosx) = sinx/cosx, which is the same as tanx.
So, now we have [tan^2x]/tanx.
And here comes the punchline: when you divide the same base with the same exponent, you simply subtract the exponents. In this case, you subtract 1 from 2, leaving you with tanx.
So the simplified form of the expression is c. tan x.
I hope that clears things up for you!
To simplify the expression [sec^2x-1]/[(sinx)(secx)], we can start by simplifying the numerator by factorizing sec^2x-1.
Starting with the numerator: sec^2x-1
Using the identity sec^2x = 1 + tan^2x, we can substitute sec^2x with (1 + tan^2x).
So now the numerator becomes: (1 + tan^2x) - 1
Simplifying further, we get: tan^2x
Now, let's simplify the denominator: (sinx)(secx)
Using the identity secx = 1/cosx, we can substitute secx with 1/cosx.
So now the denominator becomes: (sinx)(1/cosx)
Simplifying further, we get: sinx/cosx, which is equivalent to tanx.
Therefore, the expression simplifies to: (tan^2x)/(tanx).
Finally, simplifying the expression (tan^2x)/(tanx) further, we can cancel out one of the tangents, giving us the simplified expression: tanx.
So, the correct answer is (c) tanx.
To simplify the expression [sec^2x-1]/[(sinx)(secx)], we can start by simplifying each term individually.
First, let's simplify sec^2x. Recall that the Pythagorean identity for the secant function is sec^2x = 1 + tan^2x. Therefore, we can substitute sec^2x with 1 + tan^2x in the expression:
[(1 + tan^2x) - 1]/[(sinx)(secx)]
Next, let's simplify the denominator. Recall that the reciprocal of the sine function is the cosecant function, so (sinx)(secx) is equal to (1/cosx)(1/sinx), which can be written as 1/(cosxsinx):
[(1 + tan^2x) - 1]/[1/(cosxsinx)]
Now, let's simplify the numerator. By combining like terms, we have:
[tan^2x]/[1/(cosxsinx)]
To divide by a fraction, we can multiply by its reciprocal. Therefore, we can multiply the numerator by the reciprocal of the denominator:
[tan^2x] * [cosxsinx/1]
Now, let's simplify further. Notice that cosxsinx can be written as sinxcosx:
[tan^2x * sinxcosx]/1
Finally, let's simplify even more. Recall that tanx = sinx/cosx. Therefore, we can substitute tanx with sinx/cosx:
[(sinx/cosx)^2 * sinxcosx]/1
Expanding the exponent and combining like terms, we have:
[sinx^2cosx/cos^2x * sinxcosx]/1
Canceling common factors and simplifying, we get:
[sinx^3cosx/(cosx)^2]/1
The cosx terms in the numerator and denominator cancel out, leaving:
[sinx^3]/1
Therefore, the simplified form of the expression [sec^2x-1]/[(sinx)(secx)] is sinx^3.