Why sin(cosx)=(1-(1/x^2))^1/2

why sin(cosx)=(1-(1/x^2))^1/2
help pls

where does this identity comes from, i dont get it x.x

Which step is confusing you?

If it's the last one: multiply your term with (x / x) Then you will see.

why sin(secx ^-1)=(1-(1/x^2))^1/2 ... that one i dont understand that identity.

the answer is sin(cosx) +c how to eliminate de sin and cos

srry wich term ?

which step, not which problem. we know the problem you are struggling with is the identity. but what step as provided for you by wolfram are you unclear about? is it every step?

i understand the simplification but i want to know where it comes from, how do you r

it's not sin(cos(x)) it is sin(arcsec(x)). by [math] sec^-1(x) [/math] it notifies the inverse function of sec(x).

the inverse of sec is cos , i dont understand why that is the same as the Answer. what identity is that

it would help to have the actual problem rather than the solution which doesnt make sense.

the problem is easy, i dont understand this "identity"

The definition of an identity?

It is confusing as to exactly what you are asking.

where it comes from. why that is equal to that... they are telling me to use that simplification but i want to understand why

...

Sin(t) = Opposite/Hypotenuse
Sec(t) = Hypotenuse/Adjacent = x
Make a right triangle with angle t, hypotenuse = x, adjacent = 1, and opposite = sqrt(x^2 - 1^2). Sec^-1(x) = Sec^-1(x/1) = t = angle between the hypotenuse and adjacent sides. Now, when we take Sin(t), we want to take the side Opposite of the angle, and divide it by the hypotenuse. That gives us sqrt(x^2 - 1^2)/x = sqrt(x^2 - 1)/x, so, sin(sec^-1(x)) = sqrt(x^2 - 1)/x. Now, clearly this formula works for the positive values of x, but what about the negative values? For negative values, sec^-1(-x) = pi - sec^-1(x), hence, applying the trig identity for all real t, sin(pi - t) = sin(t), we get sin(sec^-1(-x)) = sin(pi - sec^-1(x)) = sin(sec^-1(x)) = sqrt(x^2 - 1)/x. However, sqrt((-x)^2 - 1)/(-x) doesn't equal sqrt(x^2 - 1)/x, so what happened? Taking a close look at the graph of arcsec(x) (which is the old term for sec^-1(x)), we see that negative values of x correspond to values in the range (pi/2, pi]. When we plug these values into sin(t), we only get non negative numbers, as sin(t) is non negative on the interval [0,pi]. This isn't consistent with our first formula, sqrt(x^2 - 1)/x, because, if we plug in a negative x, we get a negative number, which is inconsistent with sin(sec^-1(x)). we can apply our identity sin(sec^-1(-x)) = sin(sec^-1(x)), and our formula for positive x, sqrt(x^2 - 1)/x, into a single formula that is consistent with both: sqrt(x^2 -1)/abs(x).

Not the inverse of sec(x) then inverse FUNCTION of sec(x). Which means sec(arcsec(x)) = x , where [math]arcsec(x) = sec^{-1} (x) [\math]. You are confused because it as an abuse of notation.

thx for the answers, i get it know

>why that is equal to that...
Because algebra is an assigned set of rules which can be utilized to manipulate an expression into multiple forms.


>understand why
Deeper understanding comes from advanced math, abstract algebra, etc..


Think of it this way everything learned in highschool and college (barring a math degree) focuses on how to use the language of math. Like learning about grammar in english. Equally you need to be fully versed on how to read and express yourself in mathematical language. Once you master algebra and calculus/analysis then you can start reading novels as opposed to the see spot run you are working with now.

yeah im trying :)

Here's some bad proof I made for further understanding

thx thx, everything helps i have a calc 2 exam wednesday :P

das it mane

Meaning to say, dont get discouraged by the lack of why answers in Algebra and Calculus, because they aren't there. BUT, there answers to the why questions out there and with enough practice you'll be more than savy enough to wrap your head around them. Stay strong, math power.

Factor out x^2. [eqn] \sqrt { 1 - \frac { 1 } { z^2 } = \sqrt { \frac { 1 } { z^2} ( z^2 - 1 ) } [/eqn] You should be able to do the rest.

yeah, thanks :]... i know definitions are everything !