Step-by-step Solution

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

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
|◻|
=
>
<
>=
<=
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 explanation

Problem to solve:

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

Choose the solving method

1

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

$\tan\left(x\right)+\cot\left(x\right)=\frac{1}{\cos\left(x\right)}\csc\left(x\right)$
2

Multiply the fraction and term

$\tan\left(x\right)+\cot\left(x\right)=\frac{\csc\left(x\right)}{\cos\left(x\right)}$
3

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

$\tan\left(x\right)+\cot\left(x\right)=\frac{\frac{1}{\sin\left(x\right)}}{\cos\left(x\right)}$
4

Divide fractions $\frac{\frac{1}{\sin\left(x\right)}}{\cos\left(x\right)}$ with Keep, Change, Flip: $\frac{a}{b}\div c=\frac{a}{b}\div\frac{c}{1}=\frac{a}{b}\times\frac{1}{c}=\frac{a}{b\cdot c}$

$\tan\left(x\right)+\cot\left(x\right)=\frac{1}{\sin\left(x\right)\cos\left(x\right)}$
5

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

$\tan\left(x\right)+\cot\left(x\right)=\tan\left(x\right)+\cot\left(x\right)$
6

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

true

Final Answer

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

Related formulas:

3. See formulas

Time to solve it:

~ 0.04 s (SnapXam)