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:

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

Specify the solving method

1

Apply the trigonometric identity: $\displaystyle\cot(x)=\frac{\cos(x)}{\sin(x)}$

$\frac{\cos\left(x\right)}{\sin\left(x\right)}\sec\left(x\right)=\csc\left(x\right)$
2

Applying the secant identity: $\displaystyle\sec\left(\theta\right)=\frac{1}{\cos\left(\theta\right)}$

$\frac{\cos\left(x\right)}{\sin\left(x\right)}\frac{1}{\cos\left(x\right)}=\csc\left(x\right)$
3

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

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

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

$\frac{1}{\sin\left(x\right)}=\csc\left(x\right)$
5

The reciprocal sine function is cosecant: $\frac{1}{\sin(x)}=\csc(x)$

$\csc\left(x\right)=\csc\left(x\right)$
6

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

true

true

### Explore different ways to solve this problem

Prove from LHS (left-hand side)Prove from RHS (right-hand side)Express everything into Sine and Cosine
$\cot\left(x\right)\cdot\sec\left(x\right)=\csc\left(x\right)$

### Main topic:

Trigonometric Identities

~ 0.03 s