Question

Eliminate θ from the equations a sec θ – c tan θ = b and b sec θ + d tan θ = c

Answer 1

Taking sec θ = x and tan θ = y we get the equations as

ax – cy = b    .....(1)

bx + dy = c   .....(2)

(1) × d ⇒ adx – cdy = bd  ....(3)

(2) × c ⇒ bcx + cdy = c²  ......(4)

(3) + (4) ⇒ x(ad + bc) = bd + c² 

∴ x = `("bd" + "c"^2)/("ad" + "bc")`

(i.e) sec θ = `("bd"  "c")/("ad"  "bc")`

(2) × a ⇒ ax + ay = ac   ......(5)

Now (1) × b ⇒ abx – bcy = b²   ......(6)

(5) – (6) ⇒ y(ad + bc) = ac – b² 

⇒ y = `("ac" - "b"^2)/("ad" + "bc")`

(i.e) tan θ = `("ac" - "b"^2)/("ad" + "bc")`

∴ sec θ = `("bd" + "c"^2)/("ad" + "bc")` and tan θ = `("ac" + "b"^2)/("ad" + "bc")`

We know sec²θ – tan²θ = 1

⇒ `(("bd" + "c"^2)^2)/(("ad" + "bc")^2) - (("ac" - "b"^2)^2)/(("ad" + "bc")^2` = 1

⇒ `(("bd" + "c"^2)^2 - ("ac" - "b"^2)^2)/(("ad" + "bc")^2` = 1

⇒ (bd+ c²)² – (ac – b²)² = (ad +bc)²

⇒ (bd+ c²)² – (ad + bc)² = (ac +b²)²