Answer
The left side is equal to the right side
Work Step by Step
The expression on the left side $\text{cos }x+\sin x.\tan x$ can be simplified by using the quotient identity $\tan x=\frac{\sin x}{\cos x}$
$\begin{align}
& \text{cos }x+\sin x.\tan x=\frac{\cos x}{\cos x}.\cos x+\sin x.\frac{\sin x}{\cos x} \\
& =\frac{{{\cos }^{2}}x}{\cos x}+\frac{{{\sin }^{2}}x}{\cos x} \\
& =\frac{{{\cos }^{2}}x+{{\sin }^{2}}x}{\cos x}
\end{align}$
Use the Pythagorean identity, ${{\sin }^{2}}x+{{\cos }^{2}}x=1$. Therefore, the expression can be further simplified as:
$\frac{{{\cos }^{2}}x+{{\sin }^{2}}x}{\cos x}=\frac{1}{\cos x}$
As per the reciprocal identity $\frac{1}{\cos x}=\sec x$
$\begin{align}
& \frac{{{\cos }^{2}}x+{{\sin }^{2}}x}{\cos x}=\frac{1}{\cos x} \\
& =\sec x
\end{align}$
Thus, the left side is equal to the right side $\text{cos }x+\sin x.\tan x=\sec x$.
