Verify the identity 1+sinx=cos(90-x)+ cot45

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justaguide's profile pic

justaguide | College Teacher | (Level 2) Distinguished Educator

Posted on

We have to prove that 1+ sin x = cos( 90 - x) + cot 45

Let's start with cos( 90 - x) + cot 45

cos (90 - x) - cot 45

cos 45 = sin 45 = 1/sqrt 2 , cos (90 - x) = sin x

=> sin x + cot 45

=> sin x  + 1

which is the left hand side.

This proves that sin 45 + 1 = cos (90 - x) + cot 45.

giorgiana1976's profile pic

giorgiana1976 | College Teacher | (Level 3) Valedictorian

Posted on

There are 2 methods, at least to prove the identity.

One method is to manage only the left side, changing the value 1 into sin 90. Then, we'll transform teh sum into a product.

The other method is to manage only the right side and to expand the cosine of the difference.

cos (90 - x) = cos 90*cos x + sin 90*sin x

cos 90 = 0 and sin 90 = 1

cos (90 - x) = 0*cos x + 1*sin x

cos (90 - x) = sin x

cot 45 = 1

We'll substitute the terms from the right side and we'll get:

1 + sin x = sin x + 1

The addition is commutative and LHS = RHS=> the given identity,1+sinx=cos(90-x)-cot45,  is true.

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