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: $x=80°,y=10°$
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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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