Question : If $\tan(5\theta-10^\circ)=\cot(5\varphi+20^\circ)$, then the value of $(\theta+\varphi)$ is:
Option 1: $16^\circ$
Option 2: $18^\circ$
Option 3: $15^\circ$
Option 4: $20^\circ$
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Correct Answer: $16^\circ$
Solution :
$\tan(5\theta-10^\circ)=\cot(5\varphi+20^\circ)$
⇒ $\cot(90^\circ-5\theta+10^\circ)=\cot(5\varphi+20^\circ)$
⇒ $90^\circ-5\theta+10^\circ=5\varphi+20^\circ$
⇒ $100^\circ-20^\circ= 5(\theta+\varphi)$
⇒ $(\theta+\varphi)= 16^\circ$
Hence, the correct answer is $16^\circ$.
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