Question

Verify the identity.
cosθ-secθ= -sinθ tanθ

Answer 1

Step-by-step explanation:

We need to verify:

`cos\theta-sec\theta= -sin\theta tan\theta \\ \\ \\ \text{We know:} \\ \\ sec \theta=(1)/(cos \theta) \\ \\ \\ Then: \\ \\ cos\theta-(1)/(cos \theta)=(cos^2\theta-1)/(cos\theta) \\ \\ \\ Remember \ that: \\ \\ sin^2\theta +cos^2\theta=1 \therefore sin^2\theta=1-cos^2\theta \therefore -sin^2\theta=cos^2\theta -1 \\ \\ \\ Therefore: \\ \\ (cos^2\theta-1)/(cos\theta)=-(sin^2\theta)/(cos\theta) \\ \\ \\ But: \\ \\ sin^2\theta=sin\theta sin\theta`

`Then: \\ \\ -(sin^2\theta)/(cos\theta)=-(sin\theta sin\theta)/(cos\theta) \\ \\ \\ And: \\ \\ (sin\theta)/(cos\theta)=tan\theta \\ \\ \\ Finally: \\ \\ -(sin\theta sin\theta)/(cos\theta)=-sin\theta tan\theta`

Conclusion:

`cos\theta-sec\theta= -sin\theta tan\theta`