Question

If cosec theta + cot theta =p then find value of sec theta - tan theta

Answer 1

cosecθ+cotθ=p ----------------------------(1)
cosec²θ-cot²θ=1
or, (cosecθ+cotθ)(cosecθ-cotθ)=1
or, cosecθ-cotθ=1/p -----------------------(2)
Adding (1) and (2) we get,
2cosecθ=p+1/p
or, cosecθ=(p²+1)/2p
Now, cosecθ=h/p=(p²+1)/2p
By Pythagorus's theorem,
p²+b²=h²
or, b²=(p²+1)²-(2p)²
or, b²=p⁴+2p²+1-4p²
or, b²=p⁴-2p²+1
or, b²=(p²-1)²
or, b=(p²-1) (neglecting the negative sign)
∴, secθ=h/b=(p²+1)/(p²-1) and tanθ=p/b=2p/(p²-1)
∴, secθ-tanθ
=(p²+1)/(p²-1)-2p/(p²-1)
=(p²+1-2p)/(p²-1)
=(p-1)²/(p+1)(p-1)
=(p-1)/(p+1)

Answer 2

Answer:

Given :

sec A + tan A = p

I am replacing p by ' k '

sec A + tan A = k

We know :

sec A = H / B   & tan A = P / B

H / B + P / B =  k / 1

H + P / B =  k / 1

So , B = 1

H + P = k

P = k - H

From pythagoras theorem :

H² = P² + B²

H² = ( H - k )² + 1

H² = H² + k² - 2 H k + 1

2 H k = k² + 1

H = k² + 1 / 2 k

P = k - H

P = k² - 1 / 2 k

Now write k = p we have :

Base = 1

Perpendicular P = P² - 1 / 2 P

Hypotenuse H = P² + 1 / 2 P

Value of cosec A = H / P

cosec A =  P² + 1 / 2 P / P² - 1 / 2 P

cosec A = P² + 1 / P² - 1

Therefore , we got value .