Hint

Tangent has period π. It does not distribute over addition in its argument.

Worked solution

(a)

Tangent is undefined when \(\displaystyle x+ \frac{\pi }{4}= \frac{\pi }{2}+ n\pi\), so \(\displaystyle x= \frac{\pi }{4}+ n\pi\). The two exclusions are \(\displaystyle \frac{\pi }{4}\) and \(\displaystyle \frac{5\pi }{4}\).

Answer: \(\displaystyle \frac{\pi }{4}, \frac{5\pi }{4}\).

(b)

\(\displaystyle x+ \frac{\pi }{4}= - \frac{\pi }{4}+ n\pi\) gives \(\displaystyle x= - \frac{\pi }{2}+ n\pi\). The interval retains \(\displaystyle x= \frac{\pi }{2}\) and \(\displaystyle \frac{3\pi }{2}\); neither is an excluded input of this shifted function.

Answer: \(\displaystyle x= \frac{\pi }{2}, \frac{3\pi }{2}\).

(c)

The left argument is \(\displaystyle \frac{7\pi }{12}\), in quadrant II, so its tangent is negative. The right side is \(\displaystyle \sqrt{3}+1>0\). Both sides are defined and have different signs, so the proposed identity is false.

Answer: At \(\displaystyle x= \frac{\pi }{3}\), the left side is negative and the right side is positive.

Maths4U indicative marking guidance

Maths4U indicative marking guidance, not an official examination-board mark scheme. Award stated criteria once each, up to the part maximum. Accept valid equivalent methods. Follow-through applies only where explicitly stated; do not award an accuracy criterion for an incorrect final result.

(a) · 2 marks

• 1 mark: Use the cosine-zero condition for the shifted argument.
• 1 mark: Give both exclusions \(\displaystyle \frac{\pi }{4}\) and \(\displaystyle \frac{5\pi }{4}\).

Alternative methods: A correct asymptote argument is accepted.

Common error: Excluding the unshifted tangent asymptotes instead.

(b) · 2 marks

• 1 mark: Use a correct tangent family and undo the shift.
• 1 mark: Give exactly \(\displaystyle \frac{\pi }{2}\) and \(\displaystyle \frac{3\pi }{2}\).

Alternative methods: Using the family \(\displaystyle \frac{3\pi }{4}+ n\pi\) for the argument is equivalent.

Common error: Rejecting \(\displaystyle \frac{\pi }{2}\) merely because unshifted tan x is undefined there.

(c) · 3 marks

• 1 mark: Identify the left argument \(\displaystyle \frac{7\pi }{12}\) and its negative tangent sign.
• 1 mark: Show the right side \(\displaystyle \sqrt{3}+1\) is positive.
• 1 mark: Conclude the defined values differ, giving a valid counterexample.

Alternative methods: Correct evaluated numerical values at this same input are accepted.

Common error: Using an undefined input as if it supplied two comparable values.

Total: 7 marks.