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Question

Why is cos90°-x=sinx and sin90°-x=cosx?


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Solution

Prove that cos90°-x=sinx and sin90°-x=cosx.

Apply sum and difference formulas:

cosa-b=cosacosb+sinasinb...1sina-b=sinacosb-cosasinb...2

Using the first equation where a=90° and b=x we have,

cos90°-x=cos90°cosx+sin90°sinxcos90°-x=0cosx+1sinxcos90°=0,sin90°=1cos90°-x=sinx

Using the second equation where a=90° and b=x we have,

sin90°-x=sin90°cosx-cos90°sinxsin90°-x=1cosx-0sinxsin90°=1,cos90°=0sin90°-x=cosx

Hence, cos90°-x=sinx and sin90°-x=cosx.


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