Question

The value of tan[1/2cos⁻¹√5/3] is.......,Select Proper option from the given options.
(a) 2+√3/√2
(b) 3-√5/2
(c) √3-1/2√2
(d) √5+1/4

Answer 1

we have to find the value of $tan[1/2cos^{-1}\sqrt{5}/3]$

Let 1/2cos^-1(√5/3) = A

cos^-1(√5/3) = 2A

cos2A = (√5/3)

we know by formula cos2x = (1-tan²x)/(1 + tan²x)

so, cos2A = (1 - tan²A)/(1 + tan²A) = √5/3

3(1 - tan²A) = √5(1 + tan²A)

3 - √5 = (3 + √5) tan²A

tan²A = (3 - √5)(3 - √5)/(3² - √5²)

tan²A = (3 - √5)²/4

tanA = (3 - √5)/2........(1)

now, tan[1/2cos^-1(√5/3)] = tanA = (3 - √5)/2

hence, option (b) is correct.