If sinθ+cosθ=
3
11

​

​
, then what is tanθ+cotθ equal to?

Given equation:
sinθ+cosθ=
3
11

​

​

Squaring both sides:

(sinθ+cosθ)
2
=(
3
11

​

​
)
2

sin
2
θ+cos
2
θ+2sinθcosθ=
9
11
​

Using the identity sin
2
θ+cos
2
θ=1:

1+2sinθcosθ=
9
11
​

2sinθcosθ=
9
11
​
−1=
9
2
​

sinθcosθ=
9
1
​

Now, we need to find the value of tanθ+cotθ:

tanθ+cotθ=
cosθ
sinθ
​
+
sinθ
cosθ
​

tanθ+cotθ=
sinθcosθ
sin
2
θ+cos
2
θ
​

tanθ+cotθ=
sinθcosθ
1
​

Substitute the value of sinθcosθ=
9
1
​
:

tanθ+cotθ=
9
1
​

1
​
=9