Question

How do you simplify ${{\cot }^{2}}x-{{\csc }^{2}}x$ ?

Answer 1

Hint: In the given simplifying question, we can do it with a few simple steps by changing the cosec or cosecant to its reciprocal which is the sine function. Similarly, you need to convert the cotangent in the form of sine and cosine function. So, let’s see what the approach to the required question is.

Complete Step by Step Solution:
The given question which we need to simplify is ${{\cot }^{2}}x-{{\csc }^{2}}x$.
The first thing we need to do is to convert the cosec or cosecant in the form of sine, or we can say the reciprocal of sine, that means, we get,
$\Rightarrow \csc x=\dfrac{1}{\sin x}$
Similarly, we will convert the cot or cotangent in the form of sine and cosine function, that means, we get,
$\Rightarrow \cot x=\dfrac{\cos x}{\sin x}$
Now, put the required value of cosec and cot in the question, that means, we get,
$={{\left( \dfrac{\cos x}{\sin x} \right)}^{2}}-{{\left( \dfrac{1}{\sin x} \right)}^{2}}$
When we simplify it, we get,
$=\dfrac{{{\cos }^{2}}x-1}{{{\sin }^{2}}x}$
Now, we know that, ${{\cos }^{2}}x-1=-{{\sin }^{2}}x$when we apply, we get,
$=-\dfrac{{{\sin }^{2}}x}{{{\sin }^{2}}x}$
Now, when we cancel ${{\sin }^{2}}x$from the denominator and numerator, we get,
$=-1$

Therefore, after simplifying the question we get the value as -1.

Note:
There is an alternative approach for the above question ${{\cot }^{2}}x-{{\csc }^{2}}x$ .
We just need to replace ${{\csc }^{2}}x$with $1+{{\cot }^{2}}x$, which is the simple formula in trigonometric identities, that means, we get,
$={{\cot }^{2}}x-(1+{{\cot }^{2}}x)$
Removing the bracket, we get,
$={{\cot }^{2}}x-1-{{\cot }^{2}}x$
After simplifying, we get,
$=-1$
Therefore, after simplifying the question we get the value as -1.