Question

Cot (45⁰ + theta) - tan (45⁰ - theta)=what the answer ​

Answer 1

Answer:

0

Step-by-step explanation:

Given :

Cot (45⁰ + theta) - tan (45⁰ - theta)

To find :

The value of the given expression

Solution :

We know that,

$\longmapsto \mathrm {cot(45^{\circ} + \theta) = \dfrac{ cotx - 1}{cotx + 1} = \dfrac{cosx - sinx}{cosx + sinx}}$

$\longmapsto \mathrm {tan(45^{\circ} - \theta) = \dfrac{ 1 - tanx}{1 + tanx} = \dfrac{cosx - sinx}{cosx + sinx}}$

As these both values are equal they cancel out to give 0

$\implies \mathrm {\bcancel{\dfrac{cosx - sinx}{cosx + sinx}} - \bcancel{\dfrac{cosx - sinx}{cosx + sinx}}}$

$\implies \huge {\texttt{\underline{\underline{0}}}}$

❖ Extra information

$\longmapsto \mathrm {cotx -tanx }$

We know that,

$\boxed{\mathrm {cotx = \dfrac{cosx}{sinx} \ and \ tanx = \dfrac{sinx}{cosx}}}$

Substituting in the above equation, we get,

$\implies \mathrm{ \dfrac{cosx}{sinx} - \dfrac{sinx}{cosx} }$

$\implies \mathrm {\dfrac{cos^2x - sin^2x}{sinx.cosx} }$

Multiply 2 with numerator and denominator, we get,

$\implies \mathrm {\dfrac{2(cos^2x - sin^2x)}{2sinx.cosx} }$

We know that,

$\boxed { \mathrm {{cos^2x - sin^2x = cos2x \ and \ 2sinx.cosx =sin2x} }}$

So,

$\implies \mathrm {\dfrac{2cos2x }{sin2x} }$

$\implies \mathrm {2cot2x }$

So,

$\boxed {\mathrm {cotx - tanx = 2cot2x}}$

Hope it helps!!