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26 July, 02:22

Verify each identity:

1. (sin α-csc α) ² = cot²α-cos²α

2. 1-tan²A:1+tan²A=2cos²A-1

3. (cos α - sec α : sec α) + (sin α - csc α : csc α) = - 1

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  1. 26 July, 03:22
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    1.

    (sin α - csc α) ²

    = (sin α - 1/sin α) ²

    = sin²α - 2 (sin α) (1/sin α) + 1/sin²α

    = sin²α - 2 + 1/sin²α

    = (sin²α - 1) + (1/sin²α - 1)

    = - (1 - sin²α) + (1 - sin²α) / sin²α

    = - cos²α + cos²/sin²α

    = cot²α-cos²α

    2.

    (1-tan²A) / (1+tan²A)

    = (1 - sin²A / cos²A) / (1 + sin²A/cos²A)

    = (cos²A - sin²A) / (cos²A + sin²A)

    = (cos²A - (1 - cos²A)) / (1)

    = 2 cos²A - 1

    3.

    (cos α - sec α) / sec α + (sin α - csc α) / csc α

    = (cos α - sec α) / sec α + (sin α - csc α) / csc α

    = cos α/sec α - 1 + sin α / csc α - 1

    = cos α / (1 / cos α) + sin α / (1 / sin α) - 2

    = cos²α + sin²α - 2

    = 1 - 2

    = - 1
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