Question

if tan theta = a/b, find the value of cos theta + sin theta / cos theta - sin theta​

Answer 1

$\large{\underline{\underline{\red{\tt{\purple{\leadsto } GiveN:-}}}}}$

• $\sf tan\theta =\dfrac{a}{b}$

$\large{\underline{\underline{\red{\tt{\purple{\leadsto } To\:FinD:-}}}}}$

• The value of $\sf \dfrac{ cos\theta+sin\theta}{cos\theta-sin\theta}$

$\large{\underline{\underline{\red{\tt{\purple{\leadsto } FormulA\:UseD:-}}}}}$

We will use $\bf Componendo \:and\:Dividendo$

$\green{\tt If \:\:\dfrac{a}{b}\:\:=\:\:\dfrac{c}{d}}$

$\textbf{\textsf{\blue \qquad\qquad\qquad Then , }}$

$\boxed{\purple{\bf{\pink{\dag}\:\:\dfrac{a+b}{a-b}=\dfrac{c+d}{c-d}}}}$

$\large{\underline{\underline{\red{\tt{\purple{\leadsto } AnsweR:-}}}}}$

$\tt Given that :$

$\tt:\implies tan\theta=\dfrac{a}{b}$

$\tt:\implies \dfrac{sin\theta}{cos\theta}=\dfrac{a}{b}$

$\tt:\implies \dfrac{cos\theta}{sin\theta}=\dfrac{b}{a}$

$\textbf{\textsf{\pink{ \leadsto Using\:Componendo \:and\:Dividendo}}}$

$\tt:\implies\bigg\lgroup \dfrac{cos\theta+sin\theta}{cos\theta-sin\theta}\bigg\rgroup =\bigg\lgroup\dfrac{b+a}{b-a}\bigg\rgroup$

$\underline{\boxed{\red{\tt\longmapsto \dfrac{cos\theta+sin\theta}{cos\theta-sin\theta} =\dfrac{b+a}{b-a}}}}$