solve tan^-1x=cos^-1(1/√2 )
Answers
Step-by-step explanation:
Let sin
−1
10
3
=α
⇒sinα=
10
⇒cosα=
1−sin
2
α
=
1−
10
9
=
10
10−9
=
10
1
∴tanα=
cosα
sinα
=
10
1
10
3
=3
⇒α=tan
−1
3
∴α=sin
−1
10
3
=tan
−1
3
Let cos
−1
(
1+y
2
y
)=β
⇒cosβ=
1+y
2
y
⇒sinβ=
1−cos
2
β
=
1−
1+y
2
y
2
=
1+y
2
1+y
2
−y
2
=
1+y
2
1
=
1+y
2
1
∴tanβ=
cosβ
sinβ
=
1+y
2
y
1+y
2
1
=
y
1
⇒β=tan
−1
y
1
∴β=cos
−1
(
1+y
2
y
)=tan
−1
y
1
Now,tan
−1
x+cos
−1
(
1+y
2
y
)=sin
−1
10
3
⇒tan
−1
x+tan
−1
y
1
=tan
−1
3
⇒tan
−1
y
1
=tan
−1
3−tan
−1
x
We know that tan
−1
A−tan
−1
B=tan
−1
(
1+AB
A−B
)
⇒tan
−1
y
1
=tan
−1
(
1+3x
3−x
)
⇒
y
1
=
1+3x
3−x
⇒1+3x=3y−xy
⇒3x−3y=−(xy+1)
⇒3(x−y)=−(xy+1)
⇒x−y=−
3
(xy+1)
When x=1,y=2,⇒1−2=−
3
(2+1)
=−
3
3
⇒−1=−1
When x=2,y=1,⇒2−1=−
3
(2+1)
=−
3
3
⇒1
=−1
When x=3,y=2,⇒3−2=−
3
(6+1)
=−
3
7
⇒1
=
3
−7
When x=−2,y=−1,⇒−2+1=−
3
(2+1)
=−
3
3
=−1⇒−1=−1
Hence,the possible solutions are x=1,y=2 and x=−2,y=−1
∴ the positive integral solution is x=1,y=2