Solve {cos –1 x} + [tan –1 x] = 0 for real values of x. Where { . } and [ . ] are fractional part and greatest integer functions respectively.
Text Solution
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{1, cos 1}
Sol. Since – 1 ≤ x ≤ 1
∴ – 
≤ tan –1 x ≤ 
∴ [tan –1 x] = – 1, 0
When [tan –1 x] = – 1, then {cos –1 x} = 1 (not possible)
When [tan –1 x] = 0, then {cos –1 x} = 0 ∴ cos –1 x is integer
Since 0 ≤ cos –1 x ≤ π ∴ cos –1 x = 0, 1, 2, 3
x = cos 0, cos 1, cos 2, cos 3 but x ≠ cos 2, cos 3 ∴ the solution set is {1, cos 1}
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