Use identities to simplify sin²θ + tan²θ + cos²θ.

Step-by-step solution

1. 1

   Reorder to put the Pythagorean pair together. Addition is commutative, so the expression can be rearranged as

   $\left(\sin^{2}\theta+\cos^{2}\theta\right)+\tan^{2}\theta$

   Spotting the $\sin^{2}+\cos^{2}$ pair before converting anything into sines and cosines is what keeps the work to two lines.

2. 2

   Apply the fundamental identity.

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

   so the expression becomes $1+\tan^{2}\theta$.

3. 3

   Recognise the second Pythagorean identity. Dividing $\sin^{2}\theta+\cos^{2}\theta=1$ through by $\cos^{2}\theta$ gives

   $\tan^{2}\theta+1=\sec^{2}\theta$

4. 4

   State the answer.

   $\boxed{\sec^{2}\theta}$

   valid wherever $\tan\theta$ and $\sec\theta$ are defined, i.e. $\cos\theta\neq 0$.

5. 5

   Check at a convenient angle. At $\theta=60^{\circ}$: $\sin^{2}=0.75$, $\tan^{2}=3$, $\cos^{2}=0.25$, so the sum is $4$; and $\sec^{2}60^{\circ}=\left(\tfrac{1}{0.5}\right)^{2}=4$ ✓. At $\theta=45^{\circ}$: $0.5+1+0.5=2$ and $\sec^{2}45^{\circ}=\left(\sqrt2\right)^{2}=2$ ✓.