Question

Verify the identity. Justify each step.tan theta + cot theta = 1/sin theta cos theta (see image)

Answer 1

The given expression is:

`\tan\theta+\cot\theta=(1)/(\sin\theta\cos\theta)`

Rewrite the left side in terms of sin and cosine:

`\begin{gathered} \tan\theta=(\sin\theta)/(\cos\theta) \\ \cot\theta=(\cos\theta)/(\sin\theta) \\ \tan\theta+\cot\theta=(\sin\theta)/(\cos\theta)+(\cos\theta)/(\sin\theta) \end{gathered}`

Now apply the properties of fractions and solve the addition:

`\begin{gathered} (\sin\theta)/(\cos\theta)+(\cos\theta)/(\sin\theta)=(\sin\theta *\sin\theta+\cos\theta *\cos\theta)/(\sin\theta *\cos\theta) \\ =(\sin^2\theta+\cos^2\theta)/(\sin\theta *\cos\theta) \end{gathered}`

Apply the following identity:

`sin^2\theta+cos^2\theta=1`

Thus:

`(sin^2\theta+cos^2\theta)/(sin\theta cos\theta)=(1)/(\sin\theta\cos\theta)`

The verification is o.k.