Question

tan⁻¹ 1-x1+x = ½ tan⁻¹ x, (x>0)

Answer 1

Given   :  tan⁻¹((1-x)/(1+x)) = (1/2) tan⁻¹x   x > 0

To find :   समीकरण को सरल कीजिए  

Solution:

tan⁻¹((1-x)/(1+x)) = (1/2) tan⁻¹x

=> 2tan⁻¹((1-x)/(1+x)) = tan⁻¹x

दोनों तरफ Tan लेने पर

=> Tan ( 2tan⁻¹((1-x)/(1+x))  = Tan (tan⁻¹ ( x))

Tan2α  = 2Tanα/(1 - Tan²α)

α = tan⁻¹((1-x)/(1+x)

=>  2(1-x)/(1+x) /( 1  -  ((1-x)/(1+x))²) = x

= 2(1 - x)(1 + x) / ( (1 + x)² -(1 - x)² ) = x

=> 2 (1 - x²)/( 4x) = x

=> 1 - x²  = 2x²

=> 3x² = 1

=> x² = 1/3

=> x = 1/√3   ∵  x > 0

x = 1/√3

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