Question

tan(2tan⁻¹1/5-π/4)=.......,Select Proper option from the given options.
(a)14/33
(b) -7/17
(c) 17/7
(d) 24/25

Answer 1

HELLO DEAR,

tan (2tan-¹1/5 - π/4)

tan(2tan-¹1/5 - tan-¹ 1)

[as 2tan-¹x = tan-¹ {2*x/(1 - x²)} ]

tan[tan-¹(2*1/5)/{1 - (1/5)²} - tan-¹ 1]

tan[tan-¹ (2/5)/(24/25) - tan-¹1]

tan[tan-¹ 5/12 - tan-¹1]

[as tan-¹x - tan-¹y = tan-¹ {(x + y)/(1 - x*y)} ]

tan[tan-¹ {(5/12 + 1) /(1 - 5/12)}

tan[tan-¹ {(5 + 12)/12}/{(12 - 5)/12}

(17/12)/(7/12)

17/7

hence, option (c) is correct

I HOPE ITS HELP YOU,

THANKS

Answer 2

Dear Student,

Answer:Option C (17/7)

Solution:
${tan}^{ - 1} 1 = \frac{\pi}{4} \\ \\ = tan(2 {tan}^{ - 1} \frac{1}{5} - {tan}^{ - 1} 1) \\ \: \: since \: \: \: 2 {tan}^{ - 1} x = {tan}^{ - 1} ( \frac{2x}{1 - {x}^{2} } ) \\ \\ 2 {tan}^{ - 1} ( \frac{1}{5} ) = {tan}^{ - 1} ( \frac{ \frac{2}{5} }{1 - \frac{1}{25} } ) \\ \\ = {tan}^{ - 1} ( \frac{5}{12} ) \\ \\ {tan }^{ - 1} x + {tan}^{ - 1} y = {tan}^{ - 1} ( \frac{x + y}{1 - xy} ) \\ \\ = {tan }^{ - 1} \frac{5}{12} + {tan}^{ - 1} 1 = {tan}^{ - 1} ( \frac{ \frac{5}{12} + 1}{1 - \frac{5}{12} } ) \\ \\ = {tan}^{ - 1} ( \frac{17}{7} ) \\ \\ now \: \\ tan \: \: ({tan}^{ - 1} ( \frac{17}{7} )) \\ = \frac{17}{7}$
hope it helps you