Question : If $\cos x=\sin y$ and $\cot(x–40°)=\tan(50°–y)$, then the values of $x$ and $y$ are:
Option 1: $x=70°,y=2°$
Option 2: $x=75°,y=15°$
Option 3: $x=85°,y=5°$
Option 4: $x=80°,y=10°$
Correct Answer: $x=85°,y=5°$
Solution :
Given, $\cos x=\sin y$
⇒ $\cos x=\cos(90°–y)$
⇒ $x = (90°–y)$ ----------------------(equation 1)
Also, $\cot(x–40°)=\tan(50°–y)$
⇒ $\cot(x–40°)=\cot(90°–50°+y)$
⇒ $(x–40°)=(40°+y)$
⇒ $x=(80°+y)$ ------------------------(equation 2)
Now, equating the value of $x$ from equation 1 and equation 2 we get,
$(90°–y)=(80°+y)$
⇒ $2y=10°$
⇒ $y=5°$
So, $x=(90°–5°)=85°$
Hence, $x=85°,y=5°$.
Hence, the correct answer is '$x=85°,y=5°$'.
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