What is a simplified form of the expression sec^2x-1/sin x sec x ?
a)cot x
b)csc x
c)tan x
d)sec x tan x
Please help me :(
One of the basic trig identities is : tan^2 x = sec^2 x - 1
so...
(sec^2x-1)/(sin x sec x) , notice I fixed your typing
= tan^2 x /(sinx secx)
= (sin^2 x/cos^2 x)*(1/sinx)(cosx)
= sinx / cosx
= tanx
If your expression mean:
( sec² x - 1 ) / ( sin x ∙ sec x ) then:
sec² x = 1 / cos² x
sec² x - 1 = 1 / cos² x - 1 = 1 / cos² x - cos² x / cos² x =
( 1 - cos² x ) / cos² x = sin² x / cos² x = tan² x
sin x ∙ sec x = sin x ∙ 1 / cos x = sin x / cos x = tan x
( sec² x - 1 ) / ( sin x ∙ sec x ) = tan² x / tan x = tan x
Why did the expression go to therapy?
Because it had some serious sec issues! But don't worry, I'll help you simplify it.
Let's break it down step by step:
The expression is (sec^2x - 1) / (sin x * sec x).
If we factor out sec x from the numerator, we get: sec x * (sec x - 1) / (sin x * sec x).
Canceling out the sec x term in the numerator and the denominator, we are left with: (sec x - 1) / sin x.
But wait, sec x - 1 can be written as tan x, so the expression simplifies to: tan x / sin x.
And tan x / sin x is equal to cot x.
So the simplified form of the expression is a) cot x.
Now go out there and show those expressions who's boss! Keep up the good work!
To simplify the given expression, we need to apply trigonometric identities and simplify both the numerator and denominator separately.
Let's start with the numerator: sec^2(x) - 1.
Recall the identity: sec^2(x) = 1 + tan^2(x).
Therefore, we can rewrite the numerator as: 1 + tan^2(x) - 1 = tan^2(x).
Now let's simplify the denominator: sin(x) * sec(x).
Recall the identity: sec(x) = 1/cos(x).
Therefore, the denominator can be simplified as: sin(x) * (1/cos(x)) = sin(x)/cos(x) = tan(x).
Now we can rewrite the original expression as: (tan^2(x))/tan(x).
Simplifying further, we can cancel out one power of "tan" from numerator and denominator:
tan^2(x) / tan(x) = tan(x).
Therefore, the simplified form of the expression is "tan(x)".
So, the correct option is c) tan(x).