Question

If secA+tanA=5 then find the value tanA+1/tan-1 in short trick

Answer 1

When you see secA + tanA = something (let's say 'p') ... (1)
you can write, secA - tanA = 1/p ... (2)
Now add (1) and (2)
2secA = p + 1/p
secA = (p2 + 1)/2p [You can memorize this formula]
tanA = (p2 - 1)/2p

In the above question, p=2
So secA = 5/4
Now we have to find sinA. The best way to determine the value of a trigonometric function when the value of other function is given, is by making a triangle.
secϴ = Hypotenuse/Base
Here secA = 5/4, hence hypotenuse = 5 and base = 4, which means perpendicular = 3
sinA = perpendicular/hypotenuse = 3/5 = 0.6

Answer 2

secA + tanA = 5 -------(1)

we know,
sec² A - tan²A = 1
(secA -tanA)(secA + tanA) = 1
secA - tanA = 1/5 -------(2)

subtract equation (1) - (2)

2tanA = 5 - 1/5 = 24/5

tanA = 24/10 =12/5

so, ( tanA + 1)/( tanA -1) = ( 12/5 +1)/(12/5-1)
=(17)/(7) = 17/7