Prove that sec^2X cosec^2X can never be less than 2? Here ^=rise to and X=theta?

Question

Prove that sec^2X  cosec^2X can never be less than 2? Here ^=rise to and X=theta?

Answer

Answer

The question is not properly stated. However I assume it to be sec²x + cosec²x=> 1 + tan²x + 1 + cot²x=> tan²x + cot²x + 2=> (tan x + cot x)²The minimum value of the above expression occurs at x = π/4 and the minimum value is (1 + 1)² = 4But if the given question is sec²x + cos²x we can prove that the minimum value is 2.

Answer

Proof that sec^2X cosec^2X can never be less than 2

Introduction
In trigonometry, secant squared and cosecant squared are two fundamental identities that help in solving various problems. In this question, we need to prove that the product of secant squared and cosecant squared can never be less than 2.

Proof
Let us start by writing the identities of secant and cosecant in terms of sine and cosine.

- secX = 1/cosX
- cosecX = 1/sinX

Now, we can write the product of secant squared and cosecant squared as:

- (1/cos^2X) x (1/sin^2X)

Simplifying this equation, we get:

- 1/(cos^2X x sin^2X)

We know that sin^2X + cos^2X = 1 (identity of trigonometry)

Rearranging the above equation, we get:

- sin^2X = 1 - cos^2X
- cos^2X = 1 - sin^2X

Substituting the value of sin^2X in the equation of secant squared and cosecant squared, we get:

- 1/(cos^2X x (1-cos^2X))

Simplifying this equation, we get:

- 1/(cos^2X - cos^4X)

Now, let us assume that the product of secant squared and cosecant squared is less than 2. i.e.,

- (1/cos^2X) x (1/sin^2X) < />

Substituting the values of secant and cosecant in terms of sine and cosine, we get:

- (1/cos^2X) x (1/sin^2X) = (sin^2X/cos^2X) x (cos^2X/sin^2X)

Simplifying this equation, we get:

- sin^2X/cos^2X < />

Substituting the value of sin^2X in terms of cos^2X, we get:

- (1-cos^2X)/cos^2X < />

Simplifying this equation, we get:

- 1/cos^2X - 1 < />

Adding 1 to both sides of the equation, we get:

- 1/cos^2X < />

Multiplying both sides of the equation by cos^2X, we get:

- 1 < />

Dividing both sides of the equation by 3, we get:

- 1/3 < />

We know that the range of cosX is between -1 and 1. Therefore, the maximum value of cos^2X is 1. Substituting this value in the above equation, we get:

- 1/3 < />

This is a true statement. Therefore, our assumption that the product of secant squared and cosecant squared is less than 2 is wrong.

Conclusion
Hence, we have proved that the product of secant squared and cosecant squared can never be less than 2.

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