Question

if 5 cos thetha +7 thetha =7 prove that 5 sin thetha -7 cos thetha =+-5

Answer 1

Given $5\cos \theta +7 \sin \theta=7$. Now let

$x=5\sin \theta -7 \cos \theta$.

Square and add the two equations. then

$(5\cos \theta +7 \sin \theta)^2+(5\sin \theta -7 \cos \theta)^2=7^2+x^2\\ 25\cos^2 \theta +7^2 \sin^2 \theta+35\cos \theta \sin \theta+25\sin^2 \theta +7^2 \cos^2 \theta-35\cos \theta \sin \theta=7^2+x^2\\ 25(\cos^2 \theta +\sin^2 \theta)+7^2 (\sin^2 \theta+ \cos^2 \theta)+35\cos \theta \sin \theta -35\cos \theta \sin \theta=7^2+x^2\\ 25+7^2 +0=7^2+x^2\\ x^2=25\\ x=\pm 5$

The proof is complete.