Question

Solve the equation for x :
sin-'x + sin (1-x) = cos x​ see the picture

Answer 1

MySolution:: Given

sin−1(x)+sin−1(1−x)=cos−1(x)

sin−1(1−x)=cos−1(x)−sin−1(x)⇒(1−x)=sin[cos−1(x)−sin−1(x)]

Using

sin−1(x)+cos−1(x)=π2

So

(1−x)=sin[π2−2sin−1(x)]=cos(2sin−1(x))=cos(cos−1(1−2x2))

Using

2sin−1(x)=cos−1(1−2x2)

So

(1−x)=(1−2x2)⇒2x2−x=0

So

2x2−x=0⇒x=0,x=12

Now Put into Original Equation we get x=12 and

x=0 satisfy the Given equation

Answer 2

sin−1(x)+sin−1(1−x)=cos−1(x)

sin−1(1−x)=cos−1(x)−sin−1(x)⇒(1−x)=sin[cos−1(x)−sin−1(x)]

Using

sin−1(x)+cos−1(x)=π2

So

(1−x)=sin[π2−2sin−1(x)]=cos(2sin−1(x))=cos(cos−1(1−2x2))

Using

2sin−1(x)=cos−1(1−2x2)

So

(1−x)=(1−2x2)⇒2x2−x=0

So

2x2−x=0⇒x=0,x=12

now Put into Original Equation we get x=12 and

x=0 satisfy the Given equation