By doing some right triangle gymnastics, we can derive things like cos ⁡ ( arctan ⁡ x ) = 1 1 + x 2 , for x &gt; 0 cos ⁡ ( arcsin ⁡ x ) = 1 − x 2 cos ⁡ ( arcsin ⁡ x ) = 1 − x 2 tan ⁡ ( arcsin ⁡ x ) = x 1 − x 2 What about arctan ⁡ cos ⁡ ( x ), arcsin ⁡ ( tan ⁡ x ), etc?

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By doing some right triangle gymnastics, we can derive things like  cos  ⁡  (  arctan  ⁡  x  )  =      1          1      +              x        2            , for   x  &amp;gt;  0   cos  ⁡  (  arcsin  ⁡  x  )  =      1    −          x      2        cos  ⁡  (  arcsin  ⁡  x  )  =      1    −          x      2        tan  ⁡  (  arcsin  ⁡  x  )  =      x          1      −              x        2            What about   arctan  ⁡  cos  ⁡  (  x  ),   arcsin  ⁡  (  tan  ⁡  x  ), etc?

treskjeqlalo · Accepted Answer

The identities you list can all be derived by expressing the trigonometric functions in terms of each other, e.g. (glossing over sign issues)  tan  ⁡  x  =            sin      ⁡      x              cos      ⁡      x        =            sin      ⁡      x              1      −              sin        2            ⁡      x          ,and thus  tan  ⁡  (  arcsin  ⁡  x  )  =            sin      ⁡      (      arcsin      ⁡      x      )              1      −              sin        2            ⁡      (      arcsin      ⁡      x      )        =      x          1      −              x        2                .So these identities hold because there are identities between the trigonometric functions, which in a sense are due to Euler&#039;s formula.You can get similar identities in the other direction for   arcsin and   arccos because there&#039;s a suitable relationship between these two:  arccos  ⁡  x  =      π    2    −  arcsin  ⁡  x    ,and thus  arccos  ⁡  (  sin  ⁡  x  )  =      π    2    −  arcsin  ⁡  (  sin  ⁡  x  )  =      π    2    −  x    .That you can&#039;t get them for   arctan  ⁡  cos  ⁡  (  x  ) and   arcsin  ⁡  (  tan  ⁡  x  ) has to do with the fact that the identities between the corresponding inverse trigonometric functions have different arguments. For instance,  arctan  ⁡  x  =  arcsin  ⁡      x                  x        2            +      1          ,but you can&#039;t turn this into a formula for   arctan  ⁡  (  sin  ⁡  x  ) because the argument on the right-hand side isn&#039;t   x.

By doing some right triangle gymnastics, we can derive things like cos ⁡

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