If tan (A + B) = p, tan (A – B) = q, then show that .

[Hint: Use 2A = (A + B) + (A – B)]

Prove that sin 4A = 4sinA cos³A – 4cosA sin³A (corrected question)

If tan θ + sin θ = m and tan θ – sin θ = n, then prove that

m² – n² = 4 sin θ tan θ

[Hint: m + n = 2tanθ, m – n = 2 sin θ, then use m² – n² = (m + n)(m – n)]

If cosα + cosβ = 0 = sinα + sinβ, then prove that cos 2α + cos 2β = –2cos (α + β).

[Hint: cosα + cosβ)² – (sinα + sinβ)² = 0]

If then show that

[Hint: Use componendo and Dividendo]