Question

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

Answer 1

Hint : First let us evaluate the equation using the ratios from trigonometric identity, $\csc x = \dfrac{1}{{\sin x}}$ and $\cot x = \dfrac{{\cos x}}{{\sin x}}$ . Then we will further simplify this expression and hence determine the value of the term, which will be the required solution.

Complete step-by-step answer :
A reciprocal of a number is $1$ divided by that number. Another way to describe reciprocal is to point out that the product of a number and its reciprocal is $1$ .
The reciprocal identities are very useful while solving trigonometric identities. We can use it by substituting the definitions into the problem where it helps us to simplify and solve the problem. We must memorize and understand all of these identities.
The given expression that we need to simplify is ${\csc ^2}x + {\cot ^2}x$ .
Now let us rewrite the equations in terms of ratios from trigonometric identities.
We know that, $\csc x = \dfrac{1}{{\sin x}}$ and $\cot x = \dfrac{{\cos x}}{{\sin x}}$ .
Therefore, by substituting the values we have,
${\csc ^2}x + {\cot ^2}x = \dfrac{1}{{{{\sin }^2}x}} + \dfrac{{{{\cos }^2}x}}{{{{\sin }^2}x}}$
Hence, ${\csc ^2}x + {\cot ^2}x = \dfrac{{1 + {{\cos }^2}x}}{{{{\sin }^2}x}}$
So, the correct answer is “ $\dfrac{{1 + {{\cos }^2}x}}{{{{\sin }^2}x}}$ .

Note : The reciprocal relation of a trigonometric function with another trigonometric function is called reciprocal identity. Every trigonometric function has a reciprocal relation with one another trigonometric function. So, the six trigonometric ratios form six reciprocal trigonometric identities and they are used as formulas in trigonometric mathematics. The sine function is a reciprocal function of cosecant function and cosecant is also a reciprocal of sine. The cosine function is the reciprocal of secant and secant function is also a reciprocal of cosine function. Tangent function is a reciprocal of cotangent and cotangent function is also reciprocal of tangent function. The reciprocal identities are $\sin \theta = \dfrac{1}{{\csc \theta }}$ , $\cos \theta = \dfrac{1}{{\sec \theta }}$ , $\tan \theta = \dfrac{1}{{\cot \theta }}$ , $\csc \theta = \dfrac{1}{{\sin \theta }}$ , $\sec \theta = \dfrac{1}{{\cos \theta }}$ and $\cot \theta = \dfrac{1}{{\tan \theta }}$ .