Question

x=acosthta, y=bcotO prove that a^2/x^2 - b^2/y^2 =1​

Answer 1

Correct Question:-

If x = a cos θ , y = b cot θ ; prove that a²/x² - b²/y² = 1.

Answer:-

Given:-

x = a cos θ

⟹ x/a = cos θ

⟹ a/x = 1/cos θ

using sec θ = 1/cos θ we get,

⟹ a/x = sec θ

Now, Square both sides.

⟹ a²/x² = sec² θ -- equation (1).

Similarly,

y = b cot θ

⟹ y/b = cot θ

⟹ b/y = 1/cot θ

⟹ b/y = tan θ. [ ∵ tan θ = 1/cot θ ]

Squaring both sides we get,

⟹ b²/y² = tan² θ -- equation (2).

Now,

We have to prove:-

⟹ a²/x² - b²/y² = 1

Putting the respective values from equations (1) & (2) we get,

⟹ sec² θ - tan² θ = 1

using the identity sec² θ - tan² θ = 1 in LHS we get,

⟹ 1 = 1

Hence, Proved.

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Trigonometric identities:-

• sin² θ + cos² θ = 1

• sec² θ - tan² θ = 1

• cosec² θ - cot² θ = 1