Question

Based on compound angles class 11th
To prove that​

Answer 1

Proof :

L.H.S. $\displaystyle\mathrm{=\frac{sin^{4}\alpha-cos^{4}\alpha+cos^{2}\alpha}{2(1-cos\alpha)}}$

$\displaystyle\mathrm{=\frac{(sin^{2}\alpha+cos^{2}\alpha)(sin^{2}\alpha-cos^{2}\alpha)+cos^{2}\alpha}{2(1-cos\alpha)}}$

$\displaystyle\mathrm{=\frac{sin^{2}\alpha-cos^{2}\alpha+cos^{2}\alpha}{2(1-cos\alpha)}}$

$\displaystyle\mathrm{=\frac{sin^{2}\alpha}{2(1-cos\alpha)}}$

$\displaystyle\mathrm{=\frac{1-cos^{2}\alpha}{2(1-cos\alpha)}}$

$\displaystyle\mathrm{=\frac{(1+cos\alpha)(1-cos\alpha)}{2(1-cos\alpha)}}$

$\displaystyle\mathrm{=\frac{1+cos\alpha}{2}}$

$\displaystyle\mathrm{=\frac{sin^{2}\frac{\alpha}{2}+cos^{2}\frac{\alpha}{2}+cos^{2}\frac{\alpha}{2}-sin^{2}\frac{\alpha}{2}}{2}}$

$\displaystyle\mathrm{=\frac{2\:cos^{2}\frac{\alpha}{2}}{2}}$

$\displaystyle\mathrm{=cos^{2}\frac{\alpha}{2}}$

= R.H.S.

Hence, proved.