# Why does cos (90 - x) = sin (x) and sin (90 - x) = cos (x) ?

Trigonometric identities are equations that relate to different trigonometric functions and are true for all the values that lie in their domain.

## Answer: By using trigonometric identities cos(α−β)=cosα cosβ+sinα sinβ and sin(α−β)=sinα cosβ−cosα sinβ the equations cos (90 - x) = sin (x) and sin (90 - x) = cos (x) are proved.

Let us see, how to solve it.

**Explanation**:

We are using the two trigonometric identities (sum and difference formulas) shown below:

cos(α−β) = cosα cosβ + sinα sinβ——————(i)

sin(α−β) = sinα cosβ − cosα sinβ—————–(ii)

Using equation (i) where α = 90 and β =x

cos(90°−x) = cos90° cosx + sin90° sinx

On substituting cos 90° value as 0 and sin 90° as 1 we get,

cos(90°−x) = 0 cosx +1 sinx

cos(90°−x) = sinx

Using equation (ii) where α = 90° and β = x

sin(90°−x) = sin90° cosx + cos90° sinx

On substituting cos 90° value as 0 and sin 90° as 1 we get,

sin(90°−x) = 1 cosx + 0 sinx

sin(90°−x) = cosx

Hence proved

### Thus, cos(90−x) = sinx and sin(90−x) = cosx

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