If 5 secθ – 3 tanθ = 5, then what is the value of 5 tanθ – 3 secθ?

If 5 secθ – 3 tanθ = 5, then what is the value of 5 tanθ – 3 secθ? Correct Answer 3

5 sec Θ - 3tan Θ = 5

⇒ 5 √(tan² Θ + 1) = 5 + 3tanΘ

⇒ 25(tan² Θ + 1) = 25 + 9tan² Θ + 30 tanΘ

⇒ 25 tan² Θ – 9tan² Θ = 30 tanΘ

⇒ 16 tan² Θ - 30 tanΘ = 0

⇒ 2tanΘ (8 tanΘ – 15) = 0

⇒ tanΘ = 0 or tanΘ = 15/8

If tan Θ = 0     

5 sec Θ - 3 tan Θ = 5

⇒ 5sec Θ – 3 × 0 = 5

⇒ sec Θ = 1     

If tan Θ = 15/8     

5sec Θ – 3 × 15/8 = 5

⇒ 5 sec Θ = 5 + 45/8

⇒ 5 secΘ = 85 /8

⇒ sec Θ = 17/8     

So, 5 tanΘ – 3 sec Θ = 5 × 0 – 3 × 1 = -3

Or, 5 × 15/8 – 3 × 17/8 = 75/8 - 51/8 = 24/8 = 3