(cos 140^(@)-cos 100^(@))/(sin 140^(@)-sin 100^(@))=

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Use Cosine Identity: Use the cosine of the supplementary angle identity. The cosine of a supplementary angle (180° - θ) is equal to the negative cosine of the angle (cos(180° - θ) = -cos(θ)). This means that cos(140°) = -cos(40°) and cos(100°) = -cos(80°).Use Sine Identity: Use the sine of the supplementary angle identity. The sine of a supplementary angle (180° - θ) is equal to the sine of the angle (sin(180° - θ) = sin(θ)). This means that sin(140°) = sin(40°) and sin(100°) = sin(80°).Substitute Values: Substitute the values from Step 1 and Step 2 into the original expression. The expression becomes (-cos(40°) + cos(80°)) / (sin(40°) - sin(80°)).Use Co-function Identities: Use the sine and cosine co-function identities. The co-function identities state that sin(θ) = cos(90° - θ) and cos(θ) = sin(90° - θ). This means that cos(80°) = sin(10°) and cos(40°) = sin(50°).Substitute Values: Substitute the values from Step 4 into the expression from Step 3. The expression now becomes (-sin(50°) + sin(10°)) / (sin(40°) - sin(80°)).Use Sine Identities: Use the sine difference and sum identities. The sine difference identity is sin(a) - sin(b) = 2 * cos((a + b)/2) * sin((a - b)/2). Applying this to both the numerator and the denominator, we get: Numerator: 2 * cos((50° + 10°)/2) * sin((50° - 10°)/2) = 2 * cos(30°) * sin(20°) Denominator: 2 * cos((40° + 80°)/2) * sin((40° - 80°)/2) = 2 * cos(60°) * sin(-20°)Simplify Expression: Simplify the expression. Since sin(-θ) = -sin(θ), the denominator becomes -2 * cos(60°) * sin(20°). The 2's and sin(20°) terms cancel out in the numerator and denominator, leaving us with: cos(30°) / -cos(60°)Calculate Cosine Values: Calculate the exact values of cos(30°) and cos(60°). cos(30°) = √3/2 and cos(60°) = 1/2.Substitute Values: Substitute the exact values into the simplified expression. The final expression is (√3/2) / -(1/2).Simplify Final Expression: Simplify the final expression. When we divide by a fraction, it is the same as multiplying by its reciprocal. Therefore, the final answer is: (√3/2) * -(2/1) = -√3

$\frac{\cos 140^{\circ}-\cos 100^{\circ}}{\sin 140^{\circ}-\sin 100^{\circ}}=$

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cos⁡140∘−cos⁡100∘sin⁡140∘−sin⁡100∘= \frac{\cos 140^{\circ}-\cos 100^{\circ}}{\sin 140^{\circ}-\sin 100^{\circ}}= sin140∘−sin100∘cos140∘−cos100∘​=

$\frac{\cos 140^{\circ}-\cos 100^{\circ}}{\sin 140^{\circ}-\sin 100^{\circ}}=$