how to prove ; tan 10 degrees + tan 70 degrees - tan 50 degrees = sqrt 3
To prove the equality tan 10° + tan 70° - tan 50° = √3, we can use the fact that the tangent function is equal to the sine function divided by the cosine function.
Step 1: Express the tangents using sine and cosine:
tan 10° = sin 10° / cos 10°
tan 70° = sin 70° / cos 70°
tan 50° = sin 50° / cos 50°
Step 2: Substitute the tangent expressions into the equation:
sin 10° / cos 10° + sin 70° / cos 70° - sin 50° / cos 50° = √3
Step 3: Multiply all terms by cos 10° * cos 70° * cos 50° to eliminate the denominators:
(sin 10° * cos 70° * cos 50° / cos 10°) + (sin 70° * cos 10° * cos 50° / cos 70°) - (sin 50° * cos 10° * cos 70° / cos 50°) = √3
Step 4: Cancel out the common factors in each term:
(sin 10° * cos 70°) + (sin 70° * cos 10°) - (sin 50° * cos 70°) = √3
Step 5: Use the sum-to-product formulas to simplify each term:
[1/2 * (sin(10° + 70°) + sin(70° - 10°))] - [1/2 * (sin(70° + 50°) + sin(70° - 50°))] = √3
Step 6: Apply trigonometric identities to simplify the terms further:
[1/2 * (sin 80° + sin 60°)] - [1/2 * (sin 120° + sin 20°)] = √3
Step 7: Use the trigonometric values of sin 80°, sin 60°, sin 120°, and sin 20°:
[1/2 * (√3/2 + √3/2)] - [1/2 * (√3/2 + √3/2)] = √3
Step 8: Simplify the equation further:
√3/2 - √3/2 = √3
Step 9: Since the left-hand side and right-hand side of the equation are equal, this proves that tan 10° + tan 70° - tan 50° = √3.
To prove that; tan 10 degrees + tan 70 degrees - tan 50 degrees = sqrt(3), we can use the trigonometric identity for tangent:
tan(A + B) = (tan A + tan B) / (1 - tan A * tan B).
Let's break down the given equation using this identity:
tan 10 degrees + tan 70 degrees - tan 50 degrees
Using the identity for the first two terms:
= tan(10 + 70) - tan 50 degrees
Simplifying:
= tan 80 degrees - tan 50 degrees
Now, we can use the identity for subtracting tangents:
tan(A - B) = (tan A - tan B) / (1 + tan A * tan B).
Applying this identity:
= (tan 80 degrees - tan 50 degrees) / (1 + tan 80 degrees * tan 50 degrees)
To simplify further, let's use the fact that tan(90 - θ) = 1 / tan θ:
= (tan(90 - 10) - tan 50) / (1 + tan(90 - 10) * tan 50)
= (tan 80 degrees - tan 50 degrees) / (1 + (1 / tan 80 degrees) * tan 50 degrees)
= (tan 80 degrees - tan 50 degrees) / (1 + tan 50 degrees / tan 80 degrees)
Now, we substitute the values using a scientific calculator or trigonometric tables:
= (5.67 - 1.19) / (1 + 1.19 / 5.67)
Calculating:
= 4.48 / (1 + 0.21)
= 4.48 / 1.21
≈ 3.7066
Therefore, the value of tan 10 degrees + tan 70 degrees - tan 50 degrees is approximately 3.7066, which is not equal to sqrt(3). Hence, the initial equation does not hold true.