Question

If secθ+tanθ=p ,prove that = (p²-1)/(p²+1).

Answer 1

Sec∅ + tan∅ = P -------------(1)

we know,
sec²∅ - tan²∅ = 1
(sec∅ - tan∅)(sec∅ + tan∅) = 1

put , sec∅ + tan∅ = P

(sec∅ - tan∅) × P = 1

(sec∅ - tan∅) = 1/P ---------(2)

add equations (1) and (2)

2sec∅ = P + 1/P

sec∅ = ( P² + 1)/2P

cos∅ = 1/sec∅ = 2P/(P² + 1)

we know,
cos∅ = base/hypotenuse
so, perpendicular = √( h² - b²)
= √{(P²+1)² -(2P)²}
=√{P²-1)²
=(P² -1)

sin∅ =perpendicular/hypotenuse
= (p² -1)/(P² +1)

hence,
sin∅ = (P² -1)/(P² + 1)