Find the minimum value of $(\tan C – \sin A)^2 + (\cot C – \cos B)^2$ for the following given data

Let $$A, B, C$$ be real numbers such that

(i) $$(\sin A, \cos B)$$ lies on a unit circle centered at origin.

(ii) $$\tan C$$ and $$\cot C$$ are defined.

Find the minimum value of $$(\tan C – \sin A)^2 + (\cot C – \cos B)^2$$

My multiple attempts are as follows:-

Attempt $$1$$:

$$\sin^2A+\cos^2B=1$$ $$\tan^2C+\sin^2A-2\sin A\tan C+\cot^2C+\cos^2 B-2\cot C\cos B$$ $$(\tan^2C+\cot^2C)+1-2\left(\dfrac{\sin A\sin C}{\cos C}+\dfrac{\cos C\cos B}{\sin C}\right)$$

$$(\tan^2C+\cot^2C)+1-2\left(\dfrac{\sin A\sin^2 C+\cos^2 C\cos B}{\sin C\cos C}\right)$$

$$(\tan^2C+\cot^2C)+1-2\left(\dfrac{\sin^2C(\sin A-\cos B)+\cos B}{\sin C\cos C}\right)\tag{1}$$

Now from here how to proceed further.

Attempt $$2$$:

$$\sin^2A+\cos^2B=1$$ $$\sin^2A=\sin^2B$$ $$A=n\pi\pm B$$

Considering only the principal range, $$A=B$$, $$A=-B$$, $$A=n\pi-B$$, $$A=n\pi+B$$

Case $$1$$: $$A=B,A=-B$$

Put $$B=A$$ or $$B=-A$$ in equation $$(1)$$

$$(\tan^2C+\cot^2C)+1-2\left(\dfrac{\sin A\sin^2 C+\cos^2 C\cos A}{\sin C\cos C}\right)$$

$$(\tan^2C+\cot^2C)+1-2\sqrt{\sin^4C+\cos^4C}\cdot\dfrac{\sin(A+\alpha)}{\sin C\cos C}$$ $$(\tan^2C+\cot^2C)+1-2\sqrt{\tan^2C+\cot^2C}\cdot \sin(A+\alpha)$$

So minimum value will be $$3-2\sqrt{2}$$

Case $$1$$: $$A=n\pi-B,A=n\pi+B$$

Put $$B=n\pi-A$$ or $$B=A-n\pi$$

$$(\tan^2C+\cot^2C)+1-2\left(\dfrac{\sin A\sin^2 C-\cos^2 C\cos A}{\sin C\cos C}\right)$$

$$(\tan^2C+\cot^2C)+1-2\sqrt{\sin^4C+\cos^4C}\cdot\dfrac{\sin(A-\alpha)}{\sin C\cos C}$$

$$(\tan^2C+\cot^2C)+1-2\sqrt{\tan^2C+\cot^2C}\cdot\sin(A-\alpha)$$

So minimum value will be $$3-2\sqrt{2}$$

Any other way to solve this question?

1 Answer

Let $$P = (tan C, cot C)$$ lying on the curve $$xy = 1$$, $$Q = (sin A, cos B)$$lying on the unit circle $$x^2 + y^2 = 1$$ and $$O = (0, 0)$$. It's easy to prove by AM-GM inequality that $$PO \geqslant \sqrt2$$, so $$(tan C - sin A)^2 + (cot C - cos B)^2 = |PQ|^2 \geqslant (|PO| - |QO|)^2 \geqslant 3 - 2\sqrt2$$, the equality held iff $$P = (1, 1)$$ and $$Q = (\frac{\sqrt2}{2}, \frac{\sqrt2}{2})$$ or their images under O-reflection.