Question

what is the value of (cosx + sinx ) / (cosx - sinx )
1) tan(pie / 4+ x )
2) tan ( pie / 4-x )
3) tanx
4) cotx
solved it​

Answer 1

Answer:

1st is the right option

$\underline{\underline{ tan \bigg(x + \dfrac{\pi}{4}\bigg)}}$

Step-by-step explanation:

Question :

$\dfrac{cosx+sinx}{cosx-sinx}$

To Find :

The value of given expression

Solution :

Dividing the numerator and denominator by cosx in the given expression

$: \implies \dfrac{\bigg(\dfrac{cosx+sinx}{cosx} \bigg)}{\bigg(\dfrac{cosx-sinx}{cosx} \bigg)}$

$: \implies \dfrac{1 + tanx}{1 - tanx}$

We know that tan45° = 1

So,

$: \implies \dfrac{1 + tanx}{1 -(1) tanx}$

$: \implies \dfrac{tan45^{\circ} + tanx}{1 - tan45^{\circ}\times tanx}$

We know that,

$\boxed{tan (A + B ) = \dfrac{tanA + tanB}{1 - tanAtanB} }$

Here, A = 45 and B = x

So,

$: \implies \dfrac{tan45^{\circ} + tanx}{1 - tan45^{\circ}\times tanx} = tan(45 + x)$

we know that in radians 45 is also written as π/4

So,

$: \implies tan(x + 45^{\circ} ) = tan \bigg(x + \dfrac{\pi}{4}\bigg)$

Hence the final result is

$\therefore \underline{ \boxed { \mathbf{{\dfrac{cosx+sinx}{cosx-sinx} = tan \bigg(x + \dfrac{\pi}{4}\bigg) }}}}$

Hope it helps!!

Answer 2

$\sf\huge\bold{Answer:}$

Given :

$⟶ \frac{( \cos \: x+ \sin \: x ) }{ (\cos \: x - \sin \: x ) }$

To find :

Value of

$\frac{( \cos \: x+ \sin \: x ) }{ (\cos \: x - \sin \: x ) }$

Solution :

$= \frac{( \cos \: x+ \sin \: x ) }{ (\cos \: x - \sin \: x ) }$

Step 1 : Dividing cos x in both numerator and denominator

$= \frac{( \frac{ \cos \: x+ \sin \: x }{\cos \: x} )}{( \frac{ \cos \: x - \sin \: x }{\cos \: x} ) }$

Step 2 : Simplifying terms by cutting

$= \frac{ (\frac{ \cos \: x }{ \cos \: x } + \frac{ \sin \: x }{ \cos \: x } ) }{( \frac{ \cos \: x }{ \cos \: x } - \frac{ \sin \: x }{ \cos \: x })}$

We know,

$\implies \: \frac{ \sin( \alpha ) }{ \cos( \alpha ) } = \tan( \alpha )$

Step 3 : Using above identity,simplifying the term

$= \frac{1 + \tan \: x }{1 - \tan \: x }$

Similarly,

$= \frac{1 + \tan \: x }{1 - (1)(\tan \: x )}$

Step 4 : Converting the ratio to fit into tan x identity.

$\implies \: 1 = \tan(45 \degree)$

$= \frac{ \tan(45 \degree) + \tan \: x }{1 -(1)( \tan \: x) }$

$\implies \: \tan( \alpha + \beta ) = \frac{ \tan( \alpha ) + \tan( \beta ) }{1 - \tan( \alpha ) \tan( \beta ) }$

Step 5 : Inserting values in the identity,we get

$= \frac{ \tan45 \degree + \tan \: x }{1 -(\tan45 )( \tan \: x) }$

$= \tan(45 \degree + x)$

Step 6 : Converting degrees into radians

$\implies \: 45 \degree \: = \frac{\pi}{4}$

$= \tan(x + \frac{\pi}{4} )$

$\therefore$

$\fbox{Solution :}$

$\implies\frac{( \cos \: x+ \sin \: x ) }{ (\cos \: x - \sin \: x ) } = \tan(x + \frac{\pi}{4} )$