Question

Based on compund angles class 11th

To prove this:-
​

Answer 1

Proof :

L.H.S. = $\displaystyle\mathrm{\frac{sin^{4}\alpha+2\:sin\alpha\:cos\alpha-cos^{4}\alpha}{tan2\alpha-1}}$

$\displaystyle\mathrm{=\frac{(sin^{4}\alpha-cos^{4}\alpha)+2\:sin\alpha\:cos\alpha}{\frac{sin2\alpha}{cos2\alpha}-1}}$

$\displaystyle\mathrm{=\frac{(sin^{2}\alpha+cos^{2}\alpha)(sin^{2}\alpha-cos^{2}\alpha)+sin2\alpha}{\frac{sin2\alpha-cos2\alpha}{cos2\alpha}}}$

$\displaystyle\mathrm{=\frac{(sin^{2}\alpha-cos^{2}\alpha)+sin2\alpha}{\frac{sin2\alpha-cos2\alpha}{cos2\alpha}}}$

$\displaystyle\mathrm{=\frac{cos2\alpha(-cos2\alpha+sin2\alpha)}{sin2\alpha-cos2\alpha}}$

$\displaystyle\mathrm{=cos2\alpha}$ = R.H.S.

Hence, proved.

Trigonometric Rules :

  • sin²α + cos²α = 1

  • cos²α - sin²α = cos2α

  • tanα = sinα / cosα

  • 2 sinα cosα = sin2α