Prove the following trigonometry identityFor any x ∈ [ 0 , 1 ] show that arcsin ⁡ ( x ) + arccos ⁡ ( x ) = π 2

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Prove the following trigonometry identityFor any   x  ∈  [  0  ,  1  ] show that  arcsin  ⁡  (  x  )  +  arccos  ⁡  (  x  )  =      π    2

tatoas9f · Accepted Answer

As   x  ∈  [  0  ,  1  ]  ,,       0    ≤    arcsin    ⁡    x    ,    arccos    ⁡    x    ≤          π      2      If       arcsin    ⁡    x    =    A    ,    arccos    ⁡    x    =    B    ,        sin    ⁡    A    =    x        ⟹        cos    ⁡    A    =    +          1      −              x        2              ,    cos    ⁡    B    =    x        ⟹        sin    ⁡    B    =    +          1      −              x        2                  sin    ⁡    (    A    +    B    )    =    sin    ⁡    A    cos    ⁡    B    +    sin    ⁡    B    cos    ⁡    A    =    x    ⋅    x    +          1      −              x        2              ⋅          1      −              x        2              =    1  As       0    ≤    A    ,    B    ≤          π      2        ,    0    ≤    A    +    B    ≤    π    ,    A    +    B    =          π      2

Prove the following trigonometry identity For any x ∈ [ 0 , 1

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