Hint
Replace tangent by sine over cosine. In part (b), test cos x=0 before dividing by cos²x.
Worked solution
(a)
On cos x≠0, use tan x=sin x/cos x. Expanding gives (1+sin²x/cos²x)cos²x=cos²x+sin²x=1, using the Pythagorean identity.
Answer: \((1+\sin^2x/\cos^2x)\cos^2x=\cos^2x+\sin^2x=1\), for cos x≠0.
(b)
If cos x=0, the equation would be 1=0, so that case gives no solution. Otherwise divide by cos²x to get tan²x=3. Both signs tan x=±√3 give the four listed angles.
Answer: \(x=\pi/3,2\pi/3,4\pi/3,5\pi/3\).
(c)
The left side contains tangent, so the cosine-zero values π/2 and 3π/2 remain excluded. The simplified constant 1 is an extension at those points, not a value of the original expression.
Answer: \(x=\pi/2,3\pi/2\); tangent is undefined there.
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) · 3 marks
- 1 mark: Use tangent as sine divided by cosine on cos x≠0.
- 1 mark: Simplify the product to cos²x+sin²x.
- 1 mark: Conclude 1 by the Pythagorean identity with the valid domain retained.
Alternative methods: Dividing the Pythagorean identity first and then multiplying on the same domain is accepted.
Common error: Treating the expression as defined when tangent is undefined.
(b) · 3 marks
- 1 mark: Check the cosine-zero case or avoid division with an equivalent identity method.
- 1 mark: Obtain both tangent signs or an equivalent complete ratio condition.
- 1 mark: Give all four permitted angles.
Alternative methods: Using 1−cos²x=3cos²x, then cos x=±1/2, is accepted.
Common error: Taking only tan x=+√3 or silently losing a zero-denominator case.
(c) · 2 marks
- 1 mark: List π/2 and 3π/2.
- 1 mark: Explain the original tangent/domain restriction remains after simplification.
Alternative methods: An equivalent domain explanation is accepted.
Common error: Substituting an undefined tangent and then multiplying it by zero.
Total: 8 marks.