Question

How do you simplify $\sin \left( x \right)\tan \left( x \right)+\cos \left( x \right)$?

Answer 1

Hint: To simplify a given trigonometric expression, we make use of trigonometric identities that are applicable to the given expression. We start solving the problem by substituting the trigonometric quotient identity \[\tan \left( x \right) = \dfrac{\sin \left( x \right)}{\cos \left( x \right)}\] into the given expression. We then make the necessary calculations and simplify the resulting expression. To do this, we will make use of the trigonometric Pythagorean identity \[{{\sin }^{2}}\left( x \right)+{{\cos }^{2}}\left( x \right)=1\]and the quotient trigonometric identity \[\text{ }\dfrac{1}{\cos \left( x \right)}=\sec \left( x \right)\text{ }\]

Complete step-by-step answer:
From the problem, we are asked how we can simplify the given expression \[\sin \left( x \right)\tan \left( x \right)+\cos \left( x \right)\].
First of all, let \[\text{A}=\sin \left( x \right)\tan \left( x \right)+\cos \left( x \right)\,\cdots \cdots \cdots \cdots (1)\]
From the trigonometric quotient identity, we know that \[\tan \left( x \right) = \dfrac{\sin \left( x \right)}{\cos \left( x \right)}\].
Let us substitute this result into equation (1), therefore we get:
\[\Rightarrow \text{A}=\sin \left( x \right)\left( \dfrac{\sin \left( x \right)}{\cos \left( x \right)} \right)+\cos \left( x \right)\]
Now, we can simplify the expression containing the brackets by multiplying \[\sin \left( x \right)\] with \[\dfrac{\sin \left( x \right)}{\cos \left( x \right)}\] hence we get:
$\Rightarrow \text{A}=\dfrac{{{\sin }^{2}}\left( x \right)}{\cos \left( x \right)}+\cos \left( x \right)$
To simplify the expression further, let us multiply both sides of the equation with cos(x).
\[\Rightarrow \text{A}\left( \text{cos}\left( x \right) \right)={{\sin }^{2}}\left( x \right)+{{\cos }^{2}}\left( x \right)\,\cdots \cdots \cdots \cdots (2)\]
From the Pythagorean identity, we know that \[{{\sin }^{2}}(x)+{{\cos }^{2}}(x)=1\]
Let us substitute this result in equation (2), therefore we get:
$\Rightarrow \text{A}\left( \text{cos}\left( x \right) \right)=1$
Now, let us divide both sides of the equation by cos(x)
$\Rightarrow \text{A}=\dfrac{1}{\text{cos}\left( x \right)}\,\cdots \cdots \cdots \cdots (3)$
Recall the trigonometric quotient identity \[\text{ }\dfrac{1}{\cos \left( x \right)}\text{=}\sec \left( x \right)\text{ }\]
Now let’s substitute this result into equation (3), hence;
\[\Rightarrow \text{A=}\sec \left( x \right)\text{ }\]
Recall that from equation (1) $\text{A}=\sin \left( x \right)\tan \left( x \right)+\cos \left( x \right)$
$\therefore \sin \left( x \right)\tan \left( x \right)+\cos \left( x \right)\,\text{=}\sec \left( x \right)$
Hence, we have evaluated that the simplified form of the expression $\sin \left( x \right)\tan \left( x \right)+\cos \left( x \right)$ is \[\sec \left( x \right)\]

Note: Whenever we are asked to simplify a given trigonometric expression, it is important to note that the trigonometric identities and equations that are applicable to the given expression must be used. Also, when making substitutions, care must be taken so as to avoid errors. It is important to know how to prove and memorize some trigonometric identities as they come in very handy when trying to simplify some complex trigonometric expressions.