Question

please prove this.


$\frac{1 - \cos^{4}a }{ \sin^{4} \: a} = \: 1 + 2 \cot^{2} \: a$
​

Answer 1

Answer:

Step-by-step explanation:

We have,

$\dfrac{1-\cos^4(\alpha)}{\sin^4(\alpha)}$

$=\dfrac{\left(1\right)^2-\left(\cos^2(\alpha)\right)^2}{\sin^2(\alpha)\cdot\sin^2(\alpha)}$

$=\dfrac{\left(1-\cos^2(\alpha)\right)\left(1+\cos^2(\alpha)\right)}{\sin^2(\alpha)\cdot\sin^2(\alpha)}$

$=\dfrac{\sin^2(\alpha)\cdot\left(1+\cos^2(\alpha)\right)}{\sin^2(\alpha)\cdot\sin^2(\alpha)}$

$=\dfrac{1+\cos^2(\alpha)}{\sin^2(\alpha)}$

$=\dfrac{1}{\sin^2(\alpha)}+\dfrac{\cos^2(\alpha)}{\sin^2(\alpha)}$

$\tt{=cosec^2(\alpha)+cot^2(\alpha)$

$\tt{=1+cot^2(\alpha)+cot^2(\alpha)$

$\tt{=1+2\,cot^2(\alpha)$