Try NerdPal! Our new app on iOS and Android

# Prove the trigonometric identity $\cot\left(x\right)\sec\left(x\right)=\csc\left(x\right)$

Go!
Go!
1
2
3
4
5
6
7
8
9
0
a
b
c
d
f
g
m
n
u
v
w
x
y
z
.
(◻)
+
-
×
◻/◻
/
÷
2

e
π
ln
log
log
lim
d/dx
Dx
|◻|
θ
=
>
<
>=
<=
sin
cos
tan
cot
sec
csc

asin
acos
atan
acot
asec
acsc

sinh
cosh
tanh
coth
sech
csch

asinh
acosh
atanh
acoth
asech
acsch

true

## Step-by-step Solution

Problem to solve:

$\cot\left(x\right)\cdot\sec\left(x\right)=\csc\left(x\right)$

Choose the solving method

I. Express the LHS in terms of sine and cosine and simplify

1

Start from the LHS (left-hand side)

$\cot\left(x\right)\sec\left(x\right)$
2

Rewrite $\cot\left(x\right)$ in terms of sine and cosine

$\frac{\cos\left(x\right)}{\sin\left(x\right)}\sec\left(x\right)$
3

Rewrite $\sec\left(x\right)$ in terms of sine and cosine

$\frac{\cos\left(x\right)}{\sin\left(x\right)}\frac{1}{\cos\left(x\right)}$
4

Multiplying fractions $\frac{\cos\left(x\right)}{\sin\left(x\right)} \times \frac{1}{\cos\left(x\right)}$

$\frac{\cos\left(x\right)}{\sin\left(x\right)\cos\left(x\right)}$
5

Simplify the fraction $\frac{\cos\left(x\right)}{\sin\left(x\right)\cos\left(x\right)}$ by $\cos\left(x\right)$

$\frac{1}{\sin\left(x\right)}$

II. Express the RHS in terms of sine and cosine and simplify

6

Start from the RHS (right-hand side)

$\csc\left(x\right)$
7

Rewrite $\csc\left(x\right)$ in terms of sine and cosine

$\frac{1}{\sin\left(x\right)}$

III. Choose what side of the identity are we going to work on

8

To prove an identity, we usually begin to work on the side of the equality that seems to be more complicated, or the side that is not expressed in terms of sine and cosine. In this problem, we will choose to work on the left side $\frac{1}{\sin\left(x\right)}$ to reach the right side $\frac{1}{\sin\left(x\right)}$

$\frac{1}{\sin\left(x\right)}=\frac{1}{\sin\left(x\right)}$

IV. Check if we arrived at the expression we wanted to prove

9

Since both sides of the equality are equal, we have proven the identity

true

$\cot\left(x\right)\cdot\sec\left(x\right)=\csc\left(x\right)$