Answer

Verified

416.4k+ views

**Hint:**We have to make use of basic mathematics. We are going to take L.C.M. And we are also going to use some trigonometric identities. There is more than one way to do sums in trigonometry. Anyway we get the same answer. We are going to make use of ${{\sin }^{2}}x+{{\cos }^{2}}x=1$, $\dfrac{1}{\cos x}=\sec x$, and$\dfrac{1}{\sin x}=\cos ecx$ .

**Complete step-by-step solution:**

We shall first solve the left-hand side of the question.

First of all, let us take L.C.M i.e cross-multiply the denominators to each other’s numerator. It goes like this :

\[\begin{align}

& \Rightarrow \dfrac{\sin x+\cos x}{\sin x}-\dfrac{\cos x-\sin x}{\cos x} \\

& \Rightarrow \dfrac{\left( \cos x \right)\left( \cos x+\sin x \right)}{\sin x\cos x}-\dfrac{\left( \sin x \right)\left( \cos x-\sin x \right)}{\sin x\cos x} \\

& \Rightarrow \dfrac{{{\cos }^{2}}x+\cos x\sin x-\sin x\cos x+{{\sin }^{2}}x}{\sin x\cos x} \\

\end{align}\]

We can see that there are like terms with the opposite signs in the third step and they cancel out.

So, \[\cos x\sin x-\sin x\cos x\] will cancel out each other.

Now it looks like the following :

\[\Rightarrow \dfrac{{{\cos }^{2}}x+{{\sin }^{2}}x}{\sin x\cos x}\]

We know that ${{\sin }^{2}}x+{{\cos }^{2}}x=1$ . This is one of the most basic identities that we have of trigonometry. Upon applying it in the above equation, we get the following :

\[\Rightarrow \dfrac{1}{\sin x\cos x}\]

We also know that $\dfrac{1}{\cos x}=\sec x$ and $\dfrac{1}{\sin x}=\cos ecx$ . Upon applying it in the above equation we get the following :

\[\begin{align}

& \Rightarrow \dfrac{1}{\sin x\cos x} \\

& \Rightarrow \sec x\cos ecx \\

\end{align}\]

This is the result of the left-hand side of our question after simplifying.

Now let us see what we got on our right-hand side.

We got \[\sec x\cos ecx\] .

So the left-hand side is equal to the right-hand side of the question.

**Hence proved that $\dfrac{\sin x+\cos x}{\sin x}-\dfrac{\cos x-\sin x}{\cos x}=\sec x\cos ecx$ .**

**Note:**We should thoroughly remember all the trigonometric identities, functions, values of each trigonometric function, every trigonometric function’s domain, and their range. We can easily solve the questions if all these are at our fingertips.

Recently Updated Pages

what is the correct chronological order of the following class 10 social science CBSE

Which of the following was not the actual cause for class 10 social science CBSE

Which of the following statements is not correct A class 10 social science CBSE

Which of the following leaders was not present in the class 10 social science CBSE

Garampani Sanctuary is located at A Diphu Assam B Gangtok class 10 social science CBSE

Which one of the following places is not covered by class 10 social science CBSE

Trending doubts

Which are the Top 10 Largest Countries of the World?

What percentage of the solar systems mass is found class 8 physics CBSE

Fill the blanks with the suitable prepositions 1 The class 9 english CBSE

The Equation xxx + 2 is Satisfied when x is Equal to Class 10 Maths

How do you graph the function fx 4x class 9 maths CBSE

Give 10 examples for herbs , shrubs , climbers , creepers

Difference Between Plant Cell and Animal Cell

Why is there a time difference of about 5 hours between class 10 social science CBSE

Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE