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Valentin [98]
3 years ago
13

Prove the identity arcsin((x-1)/(x+1))=2arctan

sqrt{x} " alt=" \sqrt{x} " align="absmiddle" class="latex-formula">-\pi /2
Mathematics
1 answer:
kipiarov [429]3 years ago
8 0
One way would be to notice that both functions have the same derivative and then find out what the "constant" is by plugging in x=0
x
=
0
.

Here's another way. Look at
π2−2arctanx−−√=2(π4−arctanx−−√)=2(arctan1−arctanx−−√).
π
2
−
2
arctan
⁡
x
=
2
(
π
4
−
arctan
⁡
x
)
=
2
(
arctan
⁡
1
−
arctan
⁡
x
)
.
Now remember the identity for the difference of two arctangents:
arctanu−arctanv=arctanu−v1+uv.
arctan
⁡
u
−
arctan
⁡
v
=
arctan
⁡
u
−
v
1
+
u
v
.
(This follows from the usual identity for the tangent of a sum.) The left side above becomes
2arctan1−x−−√1+x−−√.
2
arctan
⁡
1
−
x
1
+
x
.
The double-angle formula for the sine says sin(2u)=2sinucosu
sin
⁡
(
2
u
)
=
2
sin
⁡
u
cos
⁡
u
. Apply that:
sin(2arctan1−x−−√1+x−−√)=2sin(arctan1−x−−√1+x−−√)cos(arctan1−x−−√1+x−−√)
sin
⁡
(
2
arctan
⁡
1
−
x
1
+
x
)
=
2
sin
⁡
(
arctan
⁡
1
−
x
1
+
x
)
cos
⁡
(
arctan
⁡
1
−
x
1
+
x
)
Now remember that sin(arctanu)=u1+u2−−−−−√
sin
⁡
(
arctan
⁡
u
)
=
u
1
+
u
2
and cos(arctanu)=11+u2−−−−−√
cos
⁡
(
arctan
⁡
u
)
=
1
1
+
u
2
Then use algebra:
2⋅(1−x√1+x√)1+(1−x√1+x√)2−−−−−−−−−−√⋅11+(1−x√1+x√)2−−−−−−−−−−√=1−x1+x.
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Oduvanchick [21]

Answer:

x = 6

Step-by-step explanation:

Given

6x² - 2x + 36 = 5x² + 10x ( subtract 5x² + 10x from both sides )

x² - 12x + 36 = 0 ← in standard form

This is a perfect square of the form

(x - a)² = x² - 2ax + a²

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ella [17]

Answer:

4 {n}^{5}  + 3.5 {n}^{5}  + ( - 2.1 {n}^{5} )

first create a bracket:

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then solve the first bracket:

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then open the remaining bracket:

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carryout the subtraction operation:

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I need help with this!
timurjin [86]

Answer:

a=11

Step-by-step explanation:

Let's solve your equation step-by-step.

√a+5=4

Solve Square Root.

√a+5=4

a+5=42(Square both sides)

a+5=16

a+5−5=16−5(Subtract 5 from both sides)

a=11

Check answers. (Plug them in to make sure they work.)

a=11(Works in original equation)

8 0
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butalik [34]

The area are in the ratio of the squares of corresponding sides.

That is  15^2 : 20^2  = 225:400

so the area of the smaller figure = (225/400) * 64

= 0.5625 * 64

= 36 cm^2  (answer)

5 0
3 years ago
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