Try NerdPal! Our new 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!
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:

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

Specify the solving method

1

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

$\frac{1}{\cos\left(x\right)}\csc\left(x\right)$
2

Applying the cosecant identity: $\displaystyle\csc\left(\theta\right)=\frac{1}{\sin\left(\theta\right)}$

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

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

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

Multiply $\frac{1}{\cos\left(x\right)\sin\left(x\right)}$ by $\frac{sin(x)^2+cos(x)^2}{sin(x)^2+cos(x)^2}$

$\frac{1}{\cos\left(x\right)\sin\left(x\right)}\frac{\sin\left(x\right)^2+\cos\left(x\right)^2}{\sin\left(x\right)^2+\cos\left(x\right)^2}$

Multiplying fractions $\frac{1}{\cos\left(x\right)\sin\left(x\right)} \times \frac{\sin\left(x\right)^2+\cos\left(x\right)^2}{\sin\left(x\right)^2+\cos\left(x\right)^2}$

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

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

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

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

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

Rewrite $\frac{1}{\cos\left(x\right)\sin\left(x\right)}$ as $\tan\left(x\right)+\cot\left(x\right)$ by applying trigonometric identities

$\tan\left(x\right)+\cot\left(x\right)$
5

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

Used formulas:

3. See formulas

Time to solve it:

~ 0.05 s