First slide

Trigonometric equations

Question

The value of $\cos ycos (\frac{π}{2} - x) - \cos (\frac{π}{2} - y) \cos x + \sin ycos (\frac{π}{2} - x) + \cos xsin (\frac{π}{2} - y)$ is zero if

Moderate

A

x= o
B

y=0
C

x=y
D

$nπ + y - \frac{π}{4} (n \in Z)$

Solution

The given expression is
cosy sinx - siny cosx + sin y sinx+ cos x cos y
sin(x - y) + cos(x -y) = 0
$\begin{array}{l} \Rightarrow \sqrt{2} (\frac{1}{\sqrt{2}} \sin (x - y) + \frac{1}{\sqrt{2}} \cos (x - y)) = 0 \\ or \sin (x - y + \frac{π}{4}) = 0 \\ \Rightarrow \frac{π}{4} + x - y = nπ, n \in Z \\ \Rightarrow x - y = nπ - \frac{π}{4}, n \in Z \\ \Rightarrow x = nπ - \frac{π}{4} + y, where n \in Z \end{array}$

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