What is the value of (sin(75)+sin(15))/(cos(105)-cos(15)) ?

The value of (sin(75)+sin(15))/(cos(105)-cos(15)) is -1

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(sin(75)+sin(15))/(cos(105)-cos(15))

\frac{sin ( 7 5 ^{\circ} ) + sin ( 1 5 ^{\circ} )}{cos ( 10 5 ^{\circ} ) - cos ( 1 5 ^{\circ} )}

- 1

Solution steps

\frac{sin ( 7 5 ^{\circ} ) + sin ( 1 5 ^{\circ} )}{cos ( 10 5 ^{\circ} ) - cos ( 1 5 ^{\circ} )}

Rewrite using trig identities:

sin (7 5^{\circ}) + sin (1 5^{\circ}) = 2 sin (4 5^{\circ}) cos (3 0^{\circ})

= \frac{2 sin ( 4 5 ^{\circ} ) cos ( 3 0 ^{\circ} )}{cos ( 10 5 ^{\circ} ) - cos ( 1 5 ^{\circ} )}

Use the following trivial identity:

sin (4 5^{\circ}) = \frac{2}{2}

Use the following trivial identity:

cos (3 0^{\circ}) = \frac{3}{2}

Rewrite using trig identities:

cos (10 5^{\circ}) = \frac{2 ( 1 - 3 )}{4}

Rewrite using trig identities:

cos (1 5^{\circ}) = \frac{6 + 2}{4}

= \frac{2 \cdot \frac{2}{2} \cdot \frac{3}{2}}{\frac{2 ( 1 - 3 )}{4} - \frac{6 + 2}{4}}

Simplify

\frac{2 \cdot \frac{2}{2} \cdot \frac{3}{2}}{\frac{2 ( 1 - 3 )}{4} - \frac{6 + 2}{4}} : - 1

= - 1

sec^{2} (8)

arccot (- 5)

cos (18 0^{\circ} - 6 0^{\circ})

cot (\frac{π}{18})

\frac{7}{cos ( 6 5 ^{\circ} )}

What is the value of (sin(75)+sin(15))/(cos(105)-cos(15)) ?
The value of (sin(75)+sin(15))/(cos(105)-cos(15)) is -1