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Q.
The equation of one of the tangents to the curve $y =\cos ( x + y ),-2 \pi \leq x \leq 2 \pi$ that is parallel to the line $x +2 y =0$, is
Application of Derivatives
Solution:
$y=\cos (x+y) \dots$(i)
$\therefore \frac{ dy }{ dx }=-\sin ( x + y )\left\{1+\frac{ dy }{ dx }\right\}$
$\therefore \frac{ dy }{ dx }=-\frac{\sin ( x + y )}{1+\sin ( x + y )}=-\frac{1}{2}$
$\Rightarrow \sin (x+y)=1$, so $\cos (x+y)=0$
$\therefore $ from (i), $y =0$ and $( x + y )=2 n \pi+\frac{\pi}{2}$
Tangent at $\left(\frac{\pi}{2}, 0\right)$ is $x +2 y =\frac{\pi}{2}$