Fantastic Equation!

Is $a=2$ the only solution to the equation below? $\boxed{a\in\mathbb{C}:\forall x\in\mathbb{R} \cos ax=a\cos^ax-1}$

#Geometry

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`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
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`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
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Comments

Yes, $a = 2$ is only solution to above equation.

As above equation is valid for all $x \in R$ , so it should also be valid for $x = 0$ . Putting $x = 0$

$\cos a(0) = a\cos^a 0 - 1$

$1 = a - 1$

$a = 2$

As only one value of $a$ comes out in solution, only one exists. Hope it helps. :)

@Zakir Husain

Aryan Sanghi - 3 months, 1 week ago

Clever solution. :)

Zakir Husain - 3 months, 1 week ago

But if $a\in (0,1)$ then multiple value of $1^a$ are there (not considering principle values)

Zakir Husain - 3 months, 1 week ago

Let $a \in (0,1)$ , then

$1 = a.1^a - 1$ $2 = a.1^a$

$1^a = \frac2a$

As $a \in (0,1)$ , RHS is real, positive and $\gt 2$ . But, LHS will be negative or $1$ or non real. So, $a \in (0,1)$ won't satisfy. Similar proof is also valid for $a \in (-1,0)$ . Hope it helps. :)

Aryan Sanghi - 3 months, 1 week ago

@Aryan Sanghi – But if $a\in\mathbb{C}-\mathbb{R}\space\space 1^a$ will have $\infty$ values.

For example : $(1)^i=(e^{in2\pi})^i=e^{-n2\pi},n\in\mathbb{Z}$ (Not considering principle values)

Zakir Husain - 3 months, 1 week ago

Very cool. The double angle formula cannot be generalized. :)

David Stiff - 3 months, 1 week ago

@Yajat Shamji @Vinayak Srivastava @Mahdi Raza @Jeff Giff @Siddharth Chakravarty @ChengYiin Ong @David Stiff @Akela Chana @Aryan Sanghi @Frisk Dreemurr

Zakir Husain - 3 months, 1 week ago

Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$