Question

How do you prove $\tan \left( x \right)\sin \left( x \right)+\cos \left( x \right)=\sec \left( x \right)$.

Answer 1

Hint: The above given question is of trigonometric identities. So, we will use the fundamental trigonometric formulas such as $\tan x=\dfrac{\sin x}{\cos x}$, $\sec x=\dfrac{1}{\cos x}$ and we will also use the identity ${{\sin }^{2}}x+{{\cos }^{2}}x=1$, to prove the above expression.

Complete step-by-step solution:
We can see that above given question is of trigonometric identity and so we will use trigonometric formulas to prove the above result $\tan \left( x \right)\sin \left( x \right)+\cos \left( x \right)=\sec \left( x \right)$.
Since, we have to prove that $\tan \left( x \right)\sin \left( x \right)+\cos \left( x \right)=\sec \left( x \right)$.
We will make the LHS term equal to the RHS.
From LHS of the equation we know that LHS = $\tan \left( x \right)\sin \left( x \right)+\cos \left( x \right)$
Since, we know that $\tan x=\dfrac{\sin x}{\cos x}$ . So, we will put $\dfrac{\sin x}{\cos x}$ in place of $\tan x$.
$\Rightarrow LHS=\tan \left( x \right)\sin \left( x \right)+\cos \left( x \right)=\dfrac{\sin x}{\cos x}\times \sin x+\cos x$
Now, we will take cos x as LCM then we will get:
$\Rightarrow \tan \left( x \right)\sin \left( x \right)+\cos \left( x \right)=\dfrac{\sin x\times \sin x+\cos x\times \cos x}{\cos x}$
$\Rightarrow \tan \left( x \right)\sin \left( x \right)+\cos \left( x \right)=\dfrac{{{\sin }^{2}}x+{{\cos }^{2}}x}{\cos x}$
Now, we will use the trigonometric identity ${{\sin }^{2}}x+{{\cos }^{2}}x=1$,
$\Rightarrow \tan \left( x \right)\sin \left( x \right)+\cos \left( x \right)=\dfrac{1}{\cos x}$
Now, we know that $\sec x=\dfrac{1}{\cos x}$, so we will put $\sec x=\dfrac{1}{\cos x}$.
$\Rightarrow \tan \left( x \right)\sin \left( x \right)+\cos \left( x \right)=\sec x$
Since, LHS = sec x which is equal to RHS.
So, LHS = RHS
Hence, proved.
This is our required solution.

Note: Students are required to note that when we are given $\sec \theta $, $\operatorname{cosec}\theta $, $\tan \theta $, and $\cot \theta $ in the trigonometric expression then we always change them into $\sin \theta $ and $\cos \theta $. Also, students are required to memorize all the trigonometric formulas otherwise they will not be able to prove the above question.