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.
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.