Question

Which expression is equal to csc x, written in terms of cos x?

Answer 1

Hey there!

The first option renders us the expression of root under 1 - cos^2(x) to trigonometric function of sin(x), so this is not the correct option, that is:

$\bf{\sqrt{1 - cos^2(x)}}$

By applying the following trigonometric identity which is, known to everyone, the addition of cos^2(x) and sin^2(x) renders "1" as the equivalence on the right hand side that is:

$\boxed{\bf{Identity \: \: is: \: cos^2(x) + sin^2(x) = 1}}$

$\bf{So, \: \: 1 - cos^2(x) = sin^2(x)}$

$\bf{\therefore \quad \sqrt{sin^2(x)}}$

Cancel the roots, via the radical rule for square roots that is:

$\bf{Radical \: \: Rule: \: \sqrt[n]{a^n} = a, \quad a \geq 0}$

$\boxed{\bf{sin(x), \quad \therefore \: \: \: First \: Option \: is \: Incorrect.}}$

Let's analyse the other options given, for this question.

Coming to Option C). If this turns out to be the required answer; this will be the final conclusion to this question.

$\bf{Expression \: \: is: \: \dfrac{1}{\sqrt{1 - cos^2(x)}}}$

Using the identity of; 1 subtracted by cos squared of variable "x" which gives us the trigonometric function of "sin(x)" that is;

$\boxed{\bf{Identity \: \: Used \: \: is: \: 1 - cos^2(x) = sin^2(x)}}$

Which in turn gives us;

$\bf{\therefore \quad \dfrac{1}{\sqrt{sin^2(x)}}}$

Now, this identity is well known; the division or the inversion of sin(x) is giving the product of "cosec(x)" or "csc(x)", and; of "sin^2(x)" as "cosec^2(x)", which is followed by a "square root" which "eliminates the radical and gives the answer", therefore our answer becomes, the final answer to this question:

$\boxed{\bf{\underline{\therefore \quad Required \: \: Answer \: \: is: \: cosec(x); \: \: \: \: OR, \: \: \: \: csc(x)}}}$

Which is the required answer or the final solution; satisfying the query.

Hope this helps you and clears your doubts on applying the trigonometric identities!!!