How to prove sin ⁡ ( π + α ) sin ⁡ ( 3 π 2 + α ) + cos ⁡ ( α − π ) cos ⁡ ( π 2 + α ) + 1 = 1 cos ⁡ α ?

Question

How to prove             sin      ⁡      (      π      +      α      )              sin      ⁡      (                        3          π                2            +      α      )        +            cos      ⁡      (      α      −      π      )              cos      ⁡      (              π        2            +      α      )      +      1        =      1          cos      ⁡      α      ?

Tianna Deleon · Accepted Answer

HINT  sin  ⁡  (            3      π        2    +  α  )  =  −  cos  ⁡  α  cos  ⁡  (      π    2    +  α  )  =  −  sin  ⁡  α  sin  ⁡  (  π  +  α  )  =  −  sin  ⁡  α  cos  ⁡  (  π  −  α  )  =  −  cos  ⁡  α

Abram Boyd · Accepted Answer

sin      ⁡      (      π      +      α      )              sin      ⁡      (                        3          π                2            +      α      )        +            cos      ⁡      (      α      −      π      )              cos      ⁡      (              π        2            +      α      )      +      1        =      1          cos      ⁡      α        sin  ⁡  (  π  +  α  )  =  −  sin  ⁡  α  sin  ⁡  (            3      π        2    +  α  )  =  −  cos  ⁡  α  cos  ⁡  (  α  −  π  )  =  cos  ⁡  (  π  −  α  )  =  −  cos  ⁡  α  cos  ⁡  (      π    2    +  α  )  =  −  sin  ⁡  αThen            sin      ⁡      (      π      +      α      )              sin      ⁡      (                        3          π                2            +      α      )        +            cos      ⁡      (      α      −      π      )              cos      ⁡      (              π        2            +      α      )      +      1        =            sin      ⁡      α              cos      ⁡      α        −            cos      ⁡      α              1      −      sin      ⁡      α        =  =            sin      ⁡      α      −              sin        2            ⁡      α      −              cos        2            ⁡      α              (      1      −      sin      ⁡      α      )      cos      ⁡      α        =  =  −            1      −      sin      ⁡      α              (      1      −      sin      ⁡      α      )      cos      ⁡      α        =  −      1          cos      ⁡      α

How to prove sin ⁡ ( π

Answered question

Answer & Explanation

New Questions in Precalculus