Question

Please give detail solution of the question ​

Answer 1

Answer:

$\blue{\boxed{\huge{ cos4 \alpha}}}$

Step-by-step explanation:

$\bold{\underline{To \: find}}\longrightarrow \: value \: of \\ \dfrac{sec8 \alpha (tan10 \alpha + tan6 \alpha )}{4(tan10 \alpha - tan6 \alpha )}$

$\bold{\underline{Concept \: used }}\longrightarrow\\ 1)\red{tan \beta = \dfrac{sin \beta }{cos \beta }} \\ 2) \green{sin( \beta + \gamma ) = sin \beta \: cos \gamma + cos \beta \: sin \gamma} \\\\ 3) \red{sin( \beta - \gamma ) = sin \beta \: cos \gamma - sin \gamma \: cos \beta}$

$4)\blue{sin2 \beta = 2 \: sin \beta \: cos \beta}$

$\bold{\underline{Solution}}\longrightarrow \: \\ \dfrac{sec8 \alpha (tan10 \alpha + tan6 \alpha )}{4(tan10 \alpha - tan6 \alpha )}$

$= \dfrac{sec8 \alpha ( \dfrac{sin10 \alpha }{cos10 \alpha } + \dfrac{sin6 \alpha }{cos6 \alpha } ) }{4( \dfrac{sin10 \alpha }{cos10 \alpha } - \dfrac{sin6 \alpha }{cos6 \alpha } ) }$

$= \dfrac{sec8 \alpha ( \dfrac{sin10 \alpha \: \: cos6 \alpha + sin6 \alpha \: \: cos10 \alpha }{cos10 \alpha \: \: cos6 \alpha } }{4( \dfrac{sin10 \alpha \: \: cos6 \alpha - cos10 \alpha \: \: sin6 \alpha }{cos10 \alpha \: \: cos6 \alpha }) }$

$cos10 \alpha \: \: cos6 \alpha \: cancel \: out \: from \: the \: fraction$

$= \dfrac{sec8 \alpha (sin10 \alpha \: \: cos6 \alpha + sin6 \alpha \: \: cos10 \alpha }{4(sin10 \alpha \: \: cos6 \alpha - cos10 \alpha \: \: sin6 \alpha )}$

$= \dfrac{sec8 \alpha \: \: sin(10 \alpha + 6 \alpha )}{4 \: sin(10 \alpha - 6 \alpha )}$

$= \dfrac{sec8 \alpha \: \: sin16 \alpha }{4 \: sin4 \alpha }$

$= \dfrac{ \dfrac{1}{cos8 \alpha } \: \: (2 \: sin8 \alpha \: cos8 \alpha )}{4 \: \: sin4 \alpha }$

$cos8 \alpha \: \: cancel \: out$

$= \dfrac{2 \: \: sin8 \alpha }{4 \: \: sin4 \alpha }$

$= \dfrac{sin8 \alpha }{2 \: \: sin4 \alpha }$

$= \dfrac{2 \: sin4 \alpha \: \: cos4 \alpha }{2 \: \: sin4 \alpha }$

$2 \: \: sin4 \alpha \: cancel \: out \: from \: numerator \: and \: denominator$

$= cos4 \alpha$