Question

Prove tanx=(1+siny)/cosy, if sin(x-y)=cosx

Answer 1

Given

• sin(x - y) = cos(x)

To Prove :

• $\sf\tan x = \dfrac {1+\sin y}{\cos y}$

Solution :

Taking the given condition :

sin (x - y) = cos(x)

$\implies\sf {\sin x \cos y - cos x \sin y = \cos x}\\\\ \sf {Dividing \: both \: \: side \: by \: \cos x }\\\\ \implies\sf {\dfrac {\sin x \cos y}{\cos x} - \dfrac {\cos x \sin y}{\cos x}= \dfrac {\cos x }{\cos x}}\\\\ \implies \sf {\tan x.\cos y - \sin y = 1 }\\\\ \implies \sf{\tan x.\cos y = 1 + \sin y}\\\\ \implies \sf{\tan x = \dfrac {1+\sin y}{\cos y}}$

$\bf {Hence \: \: Proved}$

Formula Used here :

• sin (A - B) = sinA.cosB - cosA.sinB
• tanA = sinA/cosA

Other Trigonometric Formulae

• sin(A + B) = sinAcosB + cosAsinB
• cos(A + B) = cosAcosB - sinAsiB
• cos(A - B) = cosAcosB + sinAsinB
• sinA = 1/cosecA
• cosA = 1/secA
• cotA = cosA/sinA

Answer 2

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$\huge\tt{TO~PROVE:}$

• tanx=(1+siny)/cosy, if sin(x-y)=cosx

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$\huge\tt{PROVING:}$

↪Sin x cos y - cos x sin y = cos x

↪sin x cos y /cos x - cos x sin y / cos x = cos x / cos x

↪tan x * cos y - sin y = 1

↪tan x * cos y = 1+sin y

↪tan x = 1 + sin y/cos y

Therefore, Proved.

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