Rationalised 2023-24
 28
MATHEMATICS
We now consider some examples Example 3 Show that
(i)
sin–1 (
2)
2
x1
−x
 = 2 sin–1 x, 
1
1
2
2
x
−
≤
≤
(ii)
sin–1 (
2)
2
x1
−x
 = 2 cos–1 x, 
1
1
2
≤x
≤
Solution
(i)
Let x = sin θ Then sin–1 x = θ We have
sin–1 (
2)
2
x1
−x
 = sin–1 (
)
2
2sin
1
sin
θ
−
θ
= sin–1 (2sinθ cosθ) = sin–1 (sin2θ) = 2θ
= 2 sin–1 x
(ii)
Take x = cos θ, then proceeding as above, we get, sin–1 (
2)
2
x1
−x
= 2 cos–1 x
Example 4 Express 
1
cos
tan
1
sin
x
x
−  
 
 
 
 
 
−
, 
23
2
π
π
−
<
x<
 in the simplest form