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Correct Question :-
- if tan⁻¹2x + tan⁻¹3x = π/4 .. Solve for x ?
Formula used :-
- tan⁻¹A + tan⁻¹B = tan⁻¹ [ (A + B) / ( 1 - AB) ]
- tan(π/4) = 1.
Solution :-
→ tan⁻¹2x + tan⁻¹3x = π/4
Using above formula in LHS,
→ tan⁻¹ [ (2x + 3x) / ( 1 - 2x*3x) ] = π/4
→ [ 5x / (1 - 6x²) ] = tan(π/4) = 1
→ 1 - 6x² = 5x
→ 6x² + 5x - 1 = 0
Splitting The Middle Term now,
→ 6x² + 6x - x - 1 = 0
→ 6x(x +1) - 1(x + 1) = 0
→ (x + 1)(6x - 1) = 0
Putting both Equal to Zero now,
→ x + 1 = 0
→ x = (-1) .
Or,
→ (6x - 1) = 0
→ 6x = 1
→ x = (1/6) .
Hence, value of x is (-1) & (1/6).
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