Question

How do you simplify the expression $\dfrac{\tan t+1}{\sec t}$ ?

Answer 1

Hint: In this question, we have to simplify the trigonometric function. Thus, we will use the trigonometric formulas to get the solution. First, we will convert the tan and sec function in terms of sin and cos function using the formula $\tan t=\dfrac{\sin t}{\cos t}$ and $\sec t=\dfrac{1}{\cos t}$ . After that, we will take the least common multiple in the numerator. After that, we will take the reciprocal of the denominator. In the last, we will cancel out the same terms in the denominator and the numerator, to get the required solution for the problem.

Complete step by step solution:
According to the problem, we have to simplify the given trigonometric function.
Thus, we will use the trigonometric formulas to get the solution.
The trigonometric function given to us is $\dfrac{\tan t+1}{\sec t}$ ---------- (1)
Now, we will first use the trigonometric formula $\tan t=\dfrac{\sin t}{\cos t}$ and $\sec t=\dfrac{1}{\cos t}$ in equation (1), we get
$\Rightarrow \dfrac{\dfrac{\sin t}{\cos t}+1}{\dfrac{1}{\cos t}}$
Now, we will take the least common multiple in the denominator of the above equation, we get
$\Rightarrow \dfrac{\dfrac{\sin t+\cos t}{\cos t}}{\dfrac{1}{\cos t}}$
Now, we will take the reciprocal of the denominator in the above equation using the formula \[{{\left( \dfrac{1}{a} \right)}^{-1}}=a\] , we get
$\Rightarrow \dfrac{\sin t+\cos t}{\cos t}\times \cos t$
As we know, the same terms in the division will cancel out with quotient 1, we get
$\Rightarrow \sin t+\cos t$ which is the required solution.
Therefore, for the trigonometric function $\dfrac{\tan t+1}{\sec t}$ , its simplification value is equal to $\sin t+\cos t$

Note: While solving this problem, do mention all the steps properly to avoid confusion and mathematical error. Do not write any other alphabet or symbol with trigonometric function instead of $t$ . Always mention all the formulas properly to get an accurate answer.