Question

If 1 - cosA= 1/2 show that tan²A + 4sin²A = 6​

Answer 1

`\text{Given that,}\\\\1-\cos A = \frac 12\implies \cos A = 1-\frac 12 = \frac 12\\\\\text{L.H.S}\\\\=\tan^2 A + 4 \sin^2 A\\\\=(\sin^2 A)/(\cos^2 A)+4(1-\cos^2 A)\\\\\\=(1-\cos^2 A)/(\cos^2 A) +4-4\cos^2 A\\\\\\=(1-\left( \frac 12 \right)^2 )/(\left(\frac 12 \right)^2 ) +4 -4 \left( \frac 12 \right)^2\\\\\\=(1- \frac 14)/(\frac 14)+4-4\cdot \frac 14\\\\\\=(\tfrac 34)/(\tfrac 14)+4-1\\\\\\=\frac 34 \cdot 4 +3\\\\=3+3\\\\=6\\\\=\text{R.H.S}`