The absolute value of the expression tan⁡π16+tan⁡5π16+tan⁡9π16+tan⁡13π16 is

Let θ=π16 or 8θ=π2y=tan⁡θ+tan⁡5θ+tan⁡9θ+tan⁡13θ∴ y=(tan⁡θ−cot⁡θ)+(tan⁡5θ−cot⁡5θ) [ As tan⁡13θ=tan⁡(8θ+5θ)=−cot⁡5θ and tan⁡9θ=tan⁡(8θ+θ)=−cot⁡θ]=(tan⁡θ−cot⁡θ)+(cot⁡3θ−tan⁡3θ)=sin2⁡θ−cos2⁡θsin⁡θcos⁡θ+cos2⁡3θ−sin2⁡3θsin⁡3θcos⁡3θ⇒y=2cos⁡6θsin⁡6θ−cos⁡2θsin⁡2θ=2sin⁡20cos⁡6θ−cos⁡2θsin⁡6θsin⁡6θsin⁡2θ=−2sin⁡4θcos⁡2θsin⁡2θ=−4 ∵6θ=π2−2θHence, absolute value is 4.