ðŸ‘‰ Try now NerdPal! Our new math app on iOS and Android

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

Go!
Symbolic mode
Text mode
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 

How should I solve this problem?

• Prove from LHS (left-hand side)
• Prove from RHS (right-hand side)
• Express everything into Sine and Cosine
• Exact Differential Equation
• Linear Differential Equation
• Separable Differential Equation
• Homogeneous Differential Equation
• Integrate by partial fractions
• Product of Binomials with Common Term
• FOIL Method
Can't find a method? Tell us so we can add it.
1

Starting from the left-hand side (LHS) of the identity

$\tan\left(x\right)+\cot\left(x\right)$
2

Applying the tangent identity: $\displaystyle\tan\left(\theta\right)=\frac{\sin\left(\theta\right)}{\cos\left(\theta\right)}$

$\frac{\sin\left(x\right)}{\cos\left(x\right)}+\cot\left(x\right)$
Why is tan(x) = sin(x)/cos(x) ?
3

Applying the trigonometric identity: $\cot\left(\theta \right) = \frac{\cos\left(\theta \right)}{\sin\left(\theta \right)}$

$\frac{\sin\left(x\right)}{\cos\left(x\right)}+\frac{\cos\left(x\right)}{\sin\left(x\right)}$
Why does cot(x) = cos(x)/sin(x) ?
4

The least common multiple (LCM) of a sum of algebraic fractions consists of the product of the common factors with the greatest exponent, and the uncommon factors

$L.C.M.=\cos\left(x\right)\sin\left(x\right)$
📖 Do you currently use any books to study? 🎉 Thanks for sharing!
5

Obtained the least common multiple (LCM), we place it as the denominator of each fraction, and in the numerator of each fraction we add the factors that we need to complete

$\frac{\sin\left(x\right)\sin\left(x\right)}{\cos\left(x\right)\sin\left(x\right)}+\frac{\cos\left(x\right)\cos\left(x\right)}{\cos\left(x\right)\sin\left(x\right)}$
6

Combine and simplify all terms in the same fraction with common denominator $\cos\left(x\right)\sin\left(x\right)$

$\frac{1}{\cos\left(x\right)\sin\left(x\right)}$
7

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

$\frac{\sec\left(x\right)}{\sin\left(x\right)}$
8

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

$\sec\left(x\right)\csc\left(x\right)$
9

Since we have reached the expression of our goal, we have proven the identity

true

true

##  Explore different ways to solve this problem

Solving a math problem using different methods is important because it enhances understanding, encourages critical thinking, allows for multiple solutions, and develops problem-solving strategies. Read more

###  Main Topic: Trigonometric Identities

In mathematics, trigonometric identities are equalities that involve trigonometric functions and are true for every single value of the occurring variables where both sides of the equality are defined.