👉 Try now NerdPal! Our new math app on iOS and Android

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

Step-by-step Solution

Go!
Math 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

Final Answer

true

Step-by-step Solution

Problem to solve:

$\tan\left(x\right)+\cot\left(x\right)=\sec\left(x\right)\csc\left(x\right)$

Specify the solving method

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

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

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

$L.C.M.=\cos\left(x\right)\sin\left(x\right)$
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

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

Rewrite the sum of fractions as a single fraction with the same denominator

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

When multiplying two powers that have the same base ($\sin\left(x\right)$), you can add the exponents

$\frac{\sin\left(x\right)^2+\cos\left(x\right)\cos\left(x\right)}{\cos\left(x\right)\sin\left(x\right)}$

When multiplying two powers that have the same base ($\cos\left(x\right)$), you can add the exponents

$\frac{\sin\left(x\right)^2+\cos\left(x\right)^2}{\cos\left(x\right)\sin\left(x\right)}$

Applying the pythagorean identity: $\sin^2\left(\theta\right)+\cos^2\left(\theta\right)=1$

$\frac{1}{\cos\left(x\right)\sin\left(x\right)}$
Why is sin(x)^2 + cos(x)^2 = 1 ?
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: $\displaystyle\sec\left(\theta\right)=\frac{1}{\cos\left(\theta\right)}$

$\frac{1\sec\left(x\right)}{\sin\left(x\right)}$
8

Any expression multiplied by $1$ is equal to itself

$\frac{\sec\left(x\right)}{\sin\left(x\right)}$
9

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

$\sec\left(x\right)\csc\left(x\right)$
10

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

true

Final Answer

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

Prove from RHS (right-hand side)Express everything into Sine and Cosine

Give us your feedback!

Invest in your Education!

Help us make you learn faster

Complete step-by-step math solutions. No ads.

Including multiple solving methods.

Support for more than 100 math topics.

Premium access on our iOS and Android app.

Join 500k+ students in problem solving.

Subscription. Cancel anytime.
Have a promo code?
Pay $39.97 USD securely with your payment method.
Please hold while your payment is being processed.
Create an Account
3-Month Special Plan
One-time payment of $2.97 USD.
Without automatic renewal.
Create an Account