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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