If sin alpha = a sin beta and tan alpha = b tan beta, then prove that cos^2 alpha = - Maths - Introduction to Trigonometry

Question

If sin alpha = a sin beta and tan alpha = b tan beta, then prove
that cos^2 alpha = a^2 - 1/b^2 - 1

Manbar Singh · Accepted Answer

We have,     sin α = a sin β⇒1sin β = asin α⇒cosec β = asin αNow, tan α = b tan β⇒1tan β = btan α⇒cot β = btan αNow, cosec2β - cot2β = 1⇒a2sin2α - b2tan2α = 1⇒a2sin2α - b2 cos2αsin2α = 1⇒a2 - b2 cos2α = sin2α⇒a2 - b2 cos2α = 1 - cos2α⇒a2 - 1 = b2cos2α - cos2α⇒cos2α = a2 - 1b2 - 1

If sin alpha = a sin beta and tan alpha = b tan beta, then prove that cos^2 alpha = a^2 - 1/b^2 - 1