Prove that $(sinx)^(sqrt(2)) =tan x$ ?

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Answer 1 · 2018-02-02T18:38:18

The statement is false.

We seek to prove that:

$(sinx)^(sqrt(2)) =tan x$

However, We can readily disprove that this identity using a counterexample:

Consider the case $x=pi/4$

Then,:

$LHS = (sin (pi/4))^(sqrt(2))$
$\ \ \ \ \ \ \ \ = ((sqrt2)/2)^(sqrt(2))$
$\ \ \ \ \ \ \ \ ~~ 0.6125$

However,

$RHS = tan(pi/4)$
$\ \ \ \ \ \ \ \ = 1$

As we have identified that there is a specific case $(x=pi/4)$ for which the statement is false, then the statement cannot, in general, be true.

Prove that (sinx)^(sqrt(2)) =tan x (sinx)^(sqrt(2)) =tan x ?