Is tan(x-x)=-tan(x) ?

The answer to whether tan(x-x)=-tan(x) is False

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prove tan(x-x)=-tan(x)

prove

tan (x - x) = - tan (x)

False

Solution steps

tan (x - x) = - tan (x)

Show that the two sides are not equal

For

tan (x - x) = - tan (x)

plug in

x = 1

tan (1 - 1) = 0

tan (1 - 1)

Simplify

= tan (0)

Use the following trivial identity:

tan (0) = 0

tan (0)

tan (x)

periodicity table with

πn

cycle:

x 0 \frac{π}{6} \frac{π}{4} \frac{π}{3} \frac{π}{2} \frac{2 π}{3} \frac{3 π}{4} \frac{5 π}{6} tan (x) 0 \frac{3}{3} 13 \pm \infty - 3 - 1 - \frac{3}{3}

= 0

= 0

- tan (1) = - 1.55740 \dots

- tan (1)

Refine to a decimal form

= - 1.55740 \dots

The two sides are not equal

\Rightarrow False

\frac{1 - ( sin ( x ) + cos ( x ) ) ^{2}}{sin ( 2 x )} = - 1

\frac{( cos ( x ) + 1 )}{sin ^{3} ( x )} = \frac{( csc ( x ) )}{1 - cos ( x )}

cos (x) = \frac{sin ( 2 x )}{2 sin ( x )}

\frac{1 + cot ^{2} ( x )}{sec ^{2} ( x )} = cot^{2} (x)

\frac{sec ( θ ) - 1}{1 - cos ( θ )} = \frac{1}{cos ( θ )}