Question

Use a half-angle identity to find the exact value of tan 157.5°.

Answer 1

Use the half angle identity, tan157.5=(1-cos315)/sin315
cos315=cos(-45)=√2/2, sin315=-√2/2
tan157.5=(1-√2/2)/(-√2/2)=(2-√2)/(-√2)=(2√2-2)/(-2)=1-√2

Answer 2

`tan157.5^(\circ)=-0.414`

Explanation:

We are given that
`tan157.5^(\circ)`

We have to find the exact value of
`tan157.5^(\circ)` by using a half -angle identity.

We know that

`tan(\theta)/(2)=(1-cos\theta)/(sin\theta)`

Therefore, we have

`(\theta)/(2)=157.5^(\circ)`

`\theta=157.5* 2=315^(\circ)`

Now, by using half angle identity

`\tan157.5^(\circ)=(1-cos315^(\circ))/(sin315^(\circ))`

`\tan157.5^(\circ)=(1-0.707)/(-0.707)`

`tan157.5^(\circ)=-0.414`