Question

if angle theta is divided into two parts . Tangent of one Part is K times the tangent of other and phy is the difference show that sin theta=k+1/k-1 sin phy​

Answer 1

${} \huge \bold \red{solution = > }$

$let \: \alpha \: \beta \: be \: two \: parts \: of \: angle \: theta \\ then \: given \: theta = \alpha + \beta \: and \: phi = \alpha - \beta \\ consider \: tan \alpha = k \: tan \beta \\ = \frac{tan \alpha }{tan \beta } = \frac{k}{1} \\ applying \: componendo \: and\: dividendo \\ = \frac{tan \alpha + tan \beta }{tan \alpha - tan \beta } = \frac{k + 1}{k - 1} \\ = \frac{sin \alpha \div cos \alpha + sin \ \beta \div cos \ \beta }{sin \alpha \div cos \alpha - sin \beta \div cos \beta } = \frac{k + 1}{k - 1} \\ =sin \alpha cos \beta + cos \alpha sin \beta \div cos \alpha cos \beta \div sin \alpha cos \beta - cos \alpha sin \beta \div cos \alpha cos \beta = \frac{k + 1}{k - 1} \\ we \: know \: that \: sin \: (a + - b) = sina \: cosb + - cosa \: sinb \\ sin( \alpha + \beta ) \div sin \alpha - \beta = k + 1 \div k - 1 \\ = sin \: theta \div sin \: phi \: = k + 1 \div k - 1 \\ since \: sin \: theta = \frac{k + 1}{k - 1} \times sin \: phi \\ \\ \\ hence \: proved..... \\ \\ \\ \\ {} \huge \bold \red{hope \: it \: helps \: you}$

Answer 2

Answer:

refer to the attachment for the solution.

note - here I have used lambda in place of k , so don't get confused