Question

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

Answer 1

The given expression is:

$\tan\theta+\cot\theta=\frac{1}{\sin\theta\cos\theta}$

Rewrite the left side in terms of sin and cosine:

$\begin{gathered} \tan\theta=\frac{\sin\theta}{\cos\theta} \\ \cot\theta=\frac{\cos\theta}{\sin\theta} \\ \tan\theta+\cot\theta=\frac{\sin\theta}{\cos\theta}+\frac{\cos\theta}{\sin\theta} \end{gathered}$

Now apply the properties of fractions and solve the addition:

$\begin{gathered} \frac{\sin\theta}{\cos\theta}+\frac{\cos\theta}{\sin\theta}=\frac{\sin\theta *\sin\theta+\cos\theta *\cos\theta}{\sin\theta *\cos\theta} \\ =\frac{\sin^2\theta+\cos^2\theta}{\sin\theta *\cos\theta} \end{gathered}$

Apply the following identity:

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

Thus:

$\frac{sin^2\theta+cos^2\theta}{sin\theta cos\theta}=\frac{1}{\sin\theta\cos\theta}$

The verification is o.k.