Question : If $\tanθ + \cotθ = 2$, $\theta$ is an acute angle, then find the value of $2 \tan^{25}θ + 3 \cot^{20}θ + 5 \tan^{30}θ \cot^{15}θ$.
Option 1: 8
Option 2: 6
Option 3: 10
Option 4: 12
Correct Answer: 10
Solution :
Given: The value of the trigonometric equation is $\tanθ + \cotθ = 2$ where $θ$ is an acute angle.
$\tanθ + \cotθ = 2$
⇒ $\tan 45^{\circ} + \cot45^{\circ} = 2$
⇒ $\theta = 45^{\circ}$
The value of the trigonometric expression $2 \tan^{25}45^{\circ} + 3 \cot^{20}45^{\circ} + 5 \tan^{30}45^{\circ} \cot^{15}45^{\circ}=2+3+5=10$
Hence, the correct answer is 10.
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