If \displaystyle x+y+z=\pix+y+z=π prove the trigonometric identity \displaystyle cot{\frac{x}{2}}+cot{\frac{y}{2}}+cotg\frac{z}{2}=cot{\frac{x}{2}}cot{\frac{y}{2}}cot{\frac{z}{2}}cot 2 x +cot 2 y +cotg 2 z =cot 2 x cot 2 y cot 2 z
Answers
Answer:
Hey there is some problem with your question
Answer:
x+y+z=π, so \displaystyle {\frac{x}{2}}+{\frac{y}{2}}={\frac{\pi}{2}}-{\frac{z}{2}}; {\frac{z}{2}}={\frac{\pi}{2}}-({\frac{x}{2}}+{\frac{y}{2}})
2
x
+
2
y
=
2
π
−
2
z
;
2
z
=
2
π
−(
2
x
+
2
y
). Because of \displaystyle cot\alpha={\frac{1}{tan\alpha}}cotα=
tanα
1
, we get \displaystyle cot\alpha={\frac{cos\alpha}{sin\alpha}}cotα=
sinα
cosα
. Then \displaystyle cot{\frac{x}{2}}+cot{\frac{y}{2}}+cot{\frac{z}{2}}={\frac{cos{\frac{x}{2}}}{sin{\frac{x}{2}}}}+{\frac{cos{\frac{y}{2}}}{sin{\frac{y}{2}}}}+{\frac{cos{\frac{z}{2}}}{sin{\frac{z}{2}}}}={\frac{cos{\frac{x}{2}}sin{\frac{y}{2}}+cos{\frac{y}{2}}sin{\frac{x}{2}}}{sin{\frac{x}{2}}sin{\frac{y}{2}}}}+{\frac{cos{\frac{z}{2}}}{sin{\frac{z}{2}}}}={\frac{sin({\frac{x}{2}}+{\frac{y}{2}})}{sin{\frac{x}{2}}sin{\frac{y}{2}}}}+{\frac{cos{\frac{z}{2}}}{sin{\frac{z}{2}}}}={\frac{cos{\frac{z}{2}}}{sin{\frac{x}{2}}sin{\frac{y}{2}}}}+{\frac{cos{\frac{z}{2}}}{sin{\frac{z}{2}}}}=cos{\frac{z}{2}}\cdot{\frac{sin{\frac{z}{2}}+sin{\frac{x}{2}}sin{\frac{y}{2}}}{sin{\frac{x}{2}}sin{\frac{y}{2}}sin{\frac{z}{2}}}}={\frac{cos{\frac{z}{2}}}{sin{\frac{z}{2}}}}\cdot{\frac{cos({\frac{x}{2}}+{\frac{y}{2}})+sin{\frac{x}{2}}sin{\frac{y}{2}}}{sin{\frac{x}{2}}sin{\frac{y}{2}}}}={\frac{cos{\frac{z}{2}}}{sin{\frac{z}{2}}}}\cdot{\frac{cos{\frac{x}{2}}cos{\frac{y}{2}}}{sin{\frac{x}{2}}sin{\frac{y}{2}}}}=cot{\frac{x}{2}}cot{\frac{y}{2}}cot{\frac{z}{2}}cot
2
x
+cot
2
y
+cot
2
z
=
sin
2
x
cos
2
x
+
sin
2
y
cos
2
y
+
sin
2
z
cos
2
z
=
sin
2
x
sin
2
y
cos
2
x
sin
2
y
+cos
2
y
sin
2
x
+
sin
2
z
cos
2
z
=
sin
2
x
sin
2
y
sin(
2
x
+
2
y
)
+
sin
2
z
cos
2
z
=
sin
2
x
sin
2
y
cos
2
z
+
sin
2
z
cos
2
z
=cos
2
z
⋅
sin
2
x
sin
2
y
sin
2
z
sin
2
z
+sin
2
x
sin
2
y
=
sin
2
z
cos
2
z
⋅
sin
2
x
sin
2
y
cos(
2
x
+
2
y
)+sin
2
x
sin
2
y
=
sin
2
z
cos
2
z
⋅
sin
2
x
sin
2
y
cos
2
x
cos
2
y
=cot
2
x
cot
2
y
cot
2
z
.
Step-by-step explanation: