Question

if cos a/cos b = m and cos a/ cos b show that (m2 n2) cos2 b=n2​

Answer 1

Answer:

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Step-by-step explanation:

$to \: prove = \\ (m {}^{2} + n {}^{2} ).cos {}^{2} b = n {}^{2} \\ \\ so \: here \\ \frac{cos \: a}{cos \: b} = m \\ \\ \frac{cos \: a}{sin \: b} = n \\ \\ squaring \: both \: terms \\ we \: get \\ \\ \frac{cos {}^{2} a}{cos {}^{2}b } = m {}^{2} \: \: \: \: \: \: \: \: (1)\\ \\ \frac{cos {}^{2} a}{sin {}^{2}b } = n {}^{2} \: \: \: \: \: \: \: \: \: \: (2)$

$adding \: (1) \: and \: (2) \\ we \: get \\ m {}^{2} + n {}^{2} = \frac{cos {}^{2} \: a }{cos {}^{2} \: b} + \frac{cos {}^{2} \ \: a }{sin {}^{2} \: b} \\ \\ = \frac{(cos {}^{2} \: a.sin {}^{2} \: b) +( cos {}^{2} \: a.cos {}^{2} b)}{cos { }^{2} \: b .sin {}^{2} \: b} \\ \\ = \frac{cos {}^{2} \: a(sin {}^{2} b + cos {}^{2} \: b) }{cos {}^{2} \: b.sin {}^{2} \: b } \\ \\ = \frac{cos {}^{2} \: a(1)}{cos {}^{2} \: b.sin {}^{2} \: b} \\ \\ ( m {}^{2} + n {}^{2} ).cos {}^{2} \: b = \frac{cos {}^{2} \: a }{sin {}^{2} \: b} \\ \\ (m {}^{2} + n {}^{2} ).cos {}^{2} \: b = n {}^{2} \: \: \: \: \: \: \: \: \: from \: (2) \\ \\ thus \: proved$