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

Solution:

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

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

Summary:

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.