Question

How do you prove $\dfrac{\sin x}{\csc x}+\dfrac{\cos x}{\sec x}=\sin x\csc x$?

Answer 1

Hint: In this problem we need to prove that the given equation is true. For this we need to show that value of $LHS=$value of $RHS$. So, we will consider both LHS and RHS individually. Now we will use the basic trigonometric formulas that are $\sin x=\dfrac{1}{\csc x}$, $\cos x=\dfrac{1}{\sec x}$. We will substitute those values in the LHS and simplify the equation by using the trigonometric formulas. After all simplifications we will get a value for the LHS. Now we will move to RHS and here also we will use the basic trigonometric formula $\sin x=\dfrac{1}{\csc x}$ and simplify the equation to get the value of RHS. After getting the values of LHS and RHS we will compare them to show that the given equation is true or false.

Complete step-by-step solution:
Given equation, $\dfrac{\sin x}{\csc x}+\dfrac{\cos x}{\sec x}=\sin x\csc x$.
The left-hand side part of the given equation is $\dfrac{\sin x}{\csc x}+\dfrac{\cos x}{\sec x}$ and the right-hand side part of the given equation is $\sin x\csc x$.
Considering the Left-hand side part
$LHS=\dfrac{\sin x}{\csc x}+\dfrac{\cos x}{\sec x}$
We have the basic trigonometric formulas $\sin x=\dfrac{1}{\csc x}$, $\cos x=\dfrac{1}{\sec x}$. Substituting these values in the LHS part, then we will get
$LHS=\sin x\left( \sin x \right)+\cos x\left( \cos x \right)$
We know that $a\times a={{a}^{2}}$, then the LHS is modified as
$LHS={{\sin }^{2}}x+{{\cos }^{2}}x$
We have the trigonometric identity ${{\sin }^{2}}x+{{\cos }^{2}}x=1$, then the value of LHS will be
$\therefore LHS=1$
Now considering the Right-hand side part of the given equation which is
$RHS=\sin x\csc x$
We have a basic trigonometric formula $\sin x=\dfrac{1}{\csc x}$. From this formula the value of RHS will be
$\begin{align}
  & RHS=\dfrac{\csc x}{\csc x} \\
 & \Rightarrow RHS=1 \\
\end{align}$
We have $LHS=RHS=1$. From this we can write $\dfrac{\sin x}{\csc x}+\dfrac{\cos x}{\sec x}=\sin x\csc x$. Hence proved.

Note: For this problem we have followed a simple method by using the basic trigonometric formula. There are so many methods which differ in simplification only. For example, we can multiply the term $\dfrac{\sin x}{\csc x}$ with $\dfrac{\sin x}{\sin x}$ and the term $\dfrac{\cos x}{\sec x}$ with $\dfrac{\cos x}{\cos x}$. Now we will substitute the formula $\sin x\csc x=1$, $\cos x\sec x=1$ and use the same trigonometric identity ${{\sin }^{2}}x+{{\cos }^{2}}x=1$ to get the value of LHS. For the RHS we will follow the same method.