Question

if 2costheta-sintheta=x and costheta-3sintheta=y,prove that 2x2+y2-2xy=5

Answer 1

X = 2cos@ – sin@
X² = 4cos²@ + sin²@ – 4sin@ cos@
= 1 + 3cos²@ – 4sin@ cos@
2X² = 2 + 6cos²@ – 8sin@ cos@

Y = cos@ – 3sin@
Y² = cos²@ + 9sin²@ – 6sin@ cos@
= 1 + 8sin²@ – 6sin@ cos@

XY = (2cos@ – sin@)(cos@ – 3sin@)
= 2cos²@ – 6sin@ cos@ – sin@ cos@ + 3sin²@
= 2 + sin²@ – 7cos@ sin@
2XY = 4 + 2sin²@ – 14sin@ cos@

2X² + Y² – 2XY
= 2 + 6cos²@ – 8sin@ cos@ + 1 + 8sin²@ – 6sin@ cos@ – 4 – 2sin²@ + 14sin@ cos@
= 3 + 6cos²@ + 6sin²@ –14sin@cos@ + 14sin@cos@ – 4
= 3 + 6 – 4
= 5