Let S = { x : cos − 1 x = π + sin − 1 x + sin − 1 ( 2 x + 1 ) } . Then ∑ x ∈ S ( 2 x − 1 ) 2 is equal to ______.

Question

Let S = { x : cos − 1 x = π + sin − 1 x + sin − 1 ( 2 x + 1 ) } . Then ∑ x ∈ S ​ ( 2 x − 1 ) 2 is equal to ______.

TestHub · Accepted Answer

​ cos − 1 x = π + sin − 1 x + sin − 1 ( 2 x + 1 ) 2 cos − 1 x − sin − 1 ( 2 x + 1 ) = 2 3 π ​ 2 α − β = 2 3 π ​ where cos − 1 x = α , sin − 1 ( 2 x + 1 ) = β 2 α = 2 3 π ​ + β cos 2 α = sin β 2 cos 2 α − 1 = sin β 2 x 2 − 1 = 2 x + 1 x 2 − x − 1 = 0 ​ ​ ⇒ n = 2 1 ± 5 ​ ​ = [ n = 2 1 + 5 ​ ​ rejected n = 2 1 − 5 ​ ​ ​ ∴ 4 x 2 − 4 x = 4 ( 2 x − 1 ) 2 = 5 ​

Mathematics - Inverse Trigonometric Functions Question with Solution | TestHub

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