Write the following in the simplest form: tan - 1 1 + x 2 - 1 x , x ≠ 0

Let x = tan θ Now, tan - 1 1 + x 2 - 1 x = tan - 1 1 + tan 2 θ - 1 tan θ = tan - 1 sec 2 θ - 1 tan θ = tan - 1 sec θ - 1 tan θ = tan - 1 1 - cos θ sin θ = tan - 1 2 sin 2 θ 2 2 sin θ 2 cos θ 2 = tan - 1 tan θ 2 = θ 2 Thus, the simplest form is tan - 1 x 2 .

Write the following in the simplest form: tan - 1 1 + x 2 - 1 x , x ≠ 0

Let x = tan θ Now, tan - 1 1 + x 2 - 1 x = tan - 1 1 + tan 2 θ - 1 tan θ = tan - 1 sec 2 θ - 1 tan θ = tan - 1 sec θ - 1 tan θ = tan - 1 1 - cos θ sin θ = tan - 1 2 sin 2 θ 2 2 sin θ 2 cos θ 2 = tan - 1 tan θ 2 = θ 2 Thus, the simplest form is tan - 1 x 2 .

Write the following in the simplest form: tan - 1 1 + x 2 + 1 x , x ≠ 0

Let x = tan θ Now, tan - 1 1 + x 2 + 1 x = tan - 1 1 + tan 2 θ + 1 tan θ = tan - 1 sec 2 θ + 1 tan θ = tan - 1 sec θ + 1 tan θ = tan - 1 cos θ + 1 sin θ = tan - 1 2 cos 2 θ 2 2 sin θ 2 cos θ 2 = tan - 1 cot θ 2 = tan - 1 tan π 2 - θ 2 Thus, the simplest form is π 2 - tan - 1 x 2 .

Write the following in the simplest form: tan - 1 a - x a + x , - a < x < a

Let x = a cos θ Now, tan - 1 a - x a + x = tan - 1 a - a cos θ a + a cos θ = tan - 1 1 - cos θ 1 + cos θ = tan - 1 2 sin 2 θ 2 2 cos 2 θ 2 = tan - 1 tan θ 2 = θ 2 = 1 2 cos - 1 x a

Write the following in the simplest form: tan - 1 x a + a 2 - x 2

Let x = a sin θ Now, tan - 1 x a + a 2 - x 2 = tan - 1 a sin θ a + a 2 - a 2 cos 2 θ = tan - 1 a sin θ a + a cos 2 θ = tan - 1 sin θ 1 + cos θ = tan - 1 2 sin θ 2 cos θ 2 2 cos 2 θ 2 = tan - 1 tan θ 2 = θ 2 = 1 2 sin - 1 x a

Write the following in the simplest form: sin - 1 x + 1 - x 2 2 , - 1 2 < x < 1 2

Let x = sin θ Now, sin - 1 x + 1 - x 2 2 = sin - 1 sin θ + 1 - sin 2 θ 2 = sin - 1 sin θ + cos θ 2 = sin - 1 1 2 sin θ + 1 2 cos θ = sin - 1 1 2 sin θ + 1 2 cos θ = sin - 1 sin θ + π 4 = θ + π 4 = π 4 + sin - 1 x

Write the following in the simplest form: sin - 1 1 + x + 1 - x 2 , 0 < x < 1

Let x = cos θ Now, sin - 1 1 + x + 1 - x 2 = sin - 1 1 + cos θ + 1 - cos θ 2 = sin - 1 2 cos 2 θ 2 + 2 sin 2 θ 2 2 = sin - 1 cos θ 2 + sin θ 2 2 = sin - 1 1 2 sin θ 2 + 1 2 cos θ 2 = sin - 1 sin θ 2 + π 4 = θ 2 + π 4 = cos - 1 x 2 + π 4 ∴ sin - 1 1 + x + 1 - x 2 = cos - 1 x 2 + π 4

Write the following in the simplest form: sin 2 tan - 1 1 - x 1 + x

Let x = cos θ Now, sin 2 tan - 1 1 - x 1 + x = sin 2 tan - 1 1 - cos θ 1 + cos θ = sin 2 tan - 1 2 sin 2 θ 2 2 cos 2 θ 2 = sin 2 tan - 1 tan θ 2 = sin θ = sin cos - 1 x = sin sin - 1 ( 1 - x 2 ) = 1 - x 2

Evaluate the following: sin sin - 1 7 25

Given that, sin sin - 1 7 25 . We know that, sin sin - 1 x = x , x ∈ - 1 , 1 . We can see that, 7 25 ∈ - 1 , 1 Therefore, sin sin - 1 7 25 = 7 25

Evaluate the following: sin cos - 1 5 13

sin cos - 1 5 13 = sin sin - 1 1 - 5 13 2 ∴ cos - 1 x = sin - 1 1 - x 2 = sin sin - 1 ( 1 - 25 169 ) = sin sin - 1 ( 144 169 ) = sin sin - 1 12 13 = 12 13