Question

Which of the following options is a valid proof for the equation tan 2A × Cot A - 1 = sec 2A?

A) Using the Pythagorean identity sin^2(A) + cos^2(A) = 1, prove that tan 2A × Cot A - 1 = sec 2A.

B) Using the double angle formulas for sine and cosine, prove that tan 2A × Cot A - 1 = sec 2A.

C) Using the reciprocal identities for tangent and cotangent, prove that tan 2A × Cot A - 1 = sec 2A.

D) Using the sum and difference formulas for tangent and cotangent, prove that tan 2A × Cot A - 1 = sec 2A.

Answer 1

Option B is correct because by using the double angle formulas for sine and cosine, you can manipulate the left side of the equation to match the right side, which is sec 2A.

Step-by-step explanation:

To prove the equation tan 2A × Cot A - 1 = sec 2A, we can use the double angle formulas for sine and cosine. Starting with the left side of the equation:

• tan 2A × Cot A can be rewritten as (sin 2A/cos 2A) × (cos A/sin A).
• Now, using the double angle formula, sin 2A = 2 sin A cos A and cos 2A = cos^2 A - sin^2 A, we substitute these into the expression.
• This yields (2 sin A cos A/(cos^2 A - sin^2 A)) × (cos A/sin A).
• Simplifying, we get 2 cos^2 A/(cos^2 A - sin^2 A).
• Using the Pythagorean identity sin^2 A + cos^2 A = 1, we can replace cos^2 A with 1 - sin^2 A where needed.
• The expression simplifies to 2/(1 - sin^2 A / (1 - sin^2 A)), which further simplifies to 2/(cos^2 A), which is the same as 2 cos^2 A.
• Finally, sec 2A = 1/cos 2A, and as we just proved, 2 cos^2 A = 1/(cos^2 A), satisfying the original equation.

Therefore, option B) Using the double angle formulas for sine and cosine, prove that tan 2A × Cot A - 1 = sec 2A is the valid proof.