Question

How do you simplify tan(x)cos(x)?

Answer 1

Hint: You need to know the formula tan(x) = $\dfrac{{\sin (x)}}{{\cos (x)}}$ . So, we have to use this formula in this problem to solve it. Moreover, we know that in a right-angle triangle, sin(x) = $\dfrac{{perpendicular}}{{base}}$, cos(x) = $\dfrac{{base}}{{hypotenuse}}$, and tan(x) = $\dfrac{{perpendicular}}{{base}}$ . We can also find tan(x) by dividing sin(x) with cos(x).

Complete step by step solution:
The given problem statement in which we need to simplify tan(x)cos(x).
We know that tan(x) = $\dfrac{{\sin (x)}}{{\cos (x)}}$ .
On substituting this formula in our problem stamen, we get,
$ \Rightarrow $tan(x)cos(x) = $\dfrac{{\sin (x)}}{{\cos (x)}}.\dfrac{{\cos x}}{1}$ .
After solving this, we get,
$ \Rightarrow $tan(x)cos(x)=\[\dfrac{{\sin x}}{1}\]
$ \Rightarrow $tan(x)cos(x)= \[\sin x\]

Therefore, after simplification we get tan(x)cos(x) = sin(x).

Additional Information:
There are 6 basic trigonometric functions that comprises sin, cos, tan, cosec, sec, cot as well as their inverse. Now, cosec(x) is the reciprocal of sin(x), sec(x) is the reciprocal of cos(x) and cot(x) is the reciprocal of tan(x). The formula for cosec, sec and cot are
cosec(x) = $\dfrac{{hypotenuse}}{{perpendicular}}$ , sec(x) = $\dfrac{{hypotenuse}}{{base}}$, and cot(x) = $\dfrac{{base}}{{perpendicular}}$ .
Also, we can find tan(x) by dividing sin(x) with cos(x).

Note:
There is an alternative method for solving this problem. Let’s have a look.
The given problem statement in which we need to simplify tan(x)cos(x).
Now, we know cot(x) is the reciprocal of tan(x). So, substitute tan(x)=$\dfrac{1}{{\cot (x)}}$, we get,
$ = \dfrac{1}{{\cot (x)}}.\cos (x)$
As we already know, the formula for the cot(x) is just the reciprocal of tan(x) that is a cot(x) = $\dfrac{{\cos (x)}}{{\sin (x)}}$.So, when we substitute the value of cot(x) in the above problem, we get,
$ = \dfrac{1}{{\dfrac{{\cos (x)}}{{\sin (x)}}}}.\cos (x)$
$ = \dfrac{{\sin (x)}}{{\cos (x)}}.\cos (x)$
As we can see cos(x) will cancel out from the numerator and denominator. So, after solving this we get,
$ = \sin (x)$
Therefore, after simplification we get tan(x)cos(x) = sin(x).