Question

Let cos (α + β) = \(\frac {4}{5}\) and sin (α - β) = \(\frac {5}{13}\), where 0 < α, β < \(\frac {π}{4}\) , then tan 2α=?

• A. \(\frac {20}{7}\)
• B. \(\frac {56}{33}\)
• C. \(\frac {19}{12}\)
• D. \(\frac {25}{16}\)

Answer 1

The Correct Option is B

Correct Answer: (2) 56/33

Solution:

We are given:

• cos(α + β) = 4/5
• sin(α − β) = 5/13

Step 1: Use compound angle identities:

cos(α + β) = cosα cosβ − sinα sinβ = 4/5
sin(α − β) = sinα cosβ − cosα sinβ = 5/13

Step 2: Triangle visualization

• From cos(α+β) = 4/5 ⇒ Adjacent = 4, Hypotenuse = 5 ⇒ Opposite = 3 ⇒ sin(α+β) = 3/5
• From sin(α−β) = 5/13 ⇒ Opposite = 5, Hypotenuse = 13 ⇒ Adjacent = 12 ⇒ cos(α−β) = 12/13

Step 3: Use double angle identity for sin(2α) and cos(2α):

sin(2α) = sin(α + β) cos(α − β) + cos(α + β) sin(α − β)

= (3/5 × 12/13) + (4/5 × 5/13) = 36/65 + 20/65 = 56/65

cos(2α) = cos(α + β) cos(α − β) − sin(α + β) sin(α − β)

= (4/5 × 12/13) − (3/5 × 5/13) = 48/65 − 15/65 = 33/65

Step 4: Calculate tan(2α)

tan(2α) = sin(2α) / cos(2α) = (56/65) / (33/65) = 56/33