What is the value of tan(12)-cot(78) ?

The value of tan(12)-cot(78) is 0

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tan(12)-cot(78)

tan (1 2^{\circ}) - cot (7 8^{\circ})

0

Solution steps

tan (1 2^{\circ}) - cot (7 8^{\circ})

Rewrite using trig identities:

tan (1 2^{\circ}) = cot (7 8^{\circ})

tan (1 2^{\circ})

Use the following identity:

tan (x) = cot (9 0^{\circ} - x)

cot (9 0^{\circ} - x)

Use the basic trigonometric identity:

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

= \frac{cos ( 9 0 ^{\circ} - x )}{sin ( 9 0 ^{\circ} - x )}

Use the Angle Difference identity:

cos (s - t) = cos (s) cos (t) + sin (s) sin (t)

= \frac{cos ( 9 0 ^{\circ} ) cos ( x ) + sin ( 9 0 ^{\circ} ) sin ( x )}{sin ( 9 0 ^{\circ} - x )}

Use the Angle Difference identity:

sin (s - t) = sin (s) cos (t) - cos (s) sin (t)

= \frac{cos ( 9 0 ^{\circ} ) cos ( x ) + sin ( 9 0 ^{\circ} ) sin ( x )}{sin ( 9 0 ^{\circ} ) cos ( x ) - cos ( 9 0 ^{\circ} ) sin ( x )}

Use the following trivial identity:

cos (9 0^{\circ}) = 0

Use the following trivial identity:

sin (9 0^{\circ}) = 1

= \frac{0 cos ( x ) + 1 sin ( x )}{1 cos ( x ) - 0 sin ( x )}

Simplify

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

Use the basic trigonometric identity:

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

= tan (x)

= cot (9 0^{\circ} - 1 2^{\circ})

Simplify

= cot (7 8^{\circ})

= cot (7 8^{\circ}) - cot (7 8^{\circ})

Simplify

= 0

\frac{200}{cos ( 3 0 ^{\circ} )}

arctan (\frac{2 2}{- 2 2})

\frac{20}{tan ( 3 0 ^{\circ} )}

arctan (- 0.27)

sinh (\frac{7}{20})