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Product of tan or cot over complementary angle pairs equals 1

This type of question, asked 4 times: CDS 2 2016, CDS 1 2018, CDS 2 2019, CDS 1 2020. All 4 questions below, oldest first.

Try each one first, then open its answer.

  1. Question 1

    CDS 2 2016

    85
    What is the value of tan⁡1∘tan⁡2∘tan⁡3∘tan⁡4∘…tan⁡89∘\tan 1^\circ \tan 2^\circ \tan 3^\circ \tan 4^\circ \ldots \tan 89^\circ ?

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  2. Question 2

    CDS 1 2018

    55
    What is cot⁡1∘cot⁡23∘cot⁡45∘cot⁡67∘cot⁡89∘\cot 1^\circ \cot 23^\circ \cot 45^\circ \cot 67^\circ \cot 89^\circ equal to ?

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  3. Question 3

    CDS 2 2019

    55
    What is the value of tan⁡1∘tan⁡2∘tan⁡3∘………tan⁡89∘\tan 1^\circ \tan 2^\circ \tan 3^\circ \ldots\ldots\ldots \tan 89^\circ ?

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  4. Question 4

    CDS 1 2020

    53
    Let 0<θ<90∘0 < \theta < 90^\circ and 100θ=90∘100\theta = 90^\circ. If α=∏n=199cot⁡nθ\alpha = \prod_{n=1}^{99} \cot n\theta, then which one of the following is correct?

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