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Trigonometric Identities Practice 2: Proof Techniques

Section A: Proof Moves

Question 1mcq
Simplify cosec⁡2A−cot⁡2A\cosec^2A-\cot^2A.
A.11
B.00
C.tan⁡2A\tan^2A
D.sin⁡2A\sin^2A
Question 2mcq
Which first step is useful to prove 1−cos⁡2Asin⁡2A=1\frac{1-\cos^2A}{\sin^2A}=1?
A.Replace sin⁡2A\sin^2A with cos⁡2A\cos^2A.
B.Cancel 11 from the numerator.
C.Assume A=45∘A=45^\circ only.
D.Replace 1−cos⁡2A1-\cos^2A with sin⁡2A\sin^2A.
Question 3mcq
To prove sec⁡2A−1sec⁡2A=sin⁡2A\frac{\sec^2A-1}{\sec^2A}=\sin^2A, which identity is most useful?
A.tan⁡A+cot⁡A=1\tan A+\cot A=1
B.sec⁡A=sin⁡A\sec A=\sin A
C.sec⁡2A−1=tan⁡2A\sec^2A-1=\tan^2A
D.sin⁡2A+cos⁡2A=0\sin^2A+\cos^2A=0
Question 4mcq
Which replacement correctly uses reciprocal ratios?
A.cosec⁡A=1cos⁡A\cosec A=\frac{1}{\cos A}
B.sec⁡A=1cos⁡A\sec A=\frac{1}{\cos A}
C.sec⁡A=1sin⁡A\sec A=\frac{1}{\sin A}
D.cot⁡A=1sin⁡A\cot A=\frac{1}{\sin A}
Question 5mcq
Simplify sin⁡A⋅cosec⁡A\sin A\cdot\cosec A where defined.
A.11
B.sin⁡2A\sin^2A
C.cosec⁡2A\cosec^2A
D.00

Section B: Evaluation and Validity

Question 6mcq
Which factorisation helps simplify 1−sin⁡2A1-\sin^2A?
A.Use 1−sin⁡2A=(1−sin⁡A)21-\sin^2A=(1-\sin A)^2.
B.Use 1−sin⁡2A=tan⁡2A1-\sin^2A=\tan^2A.
C.Use 1−sin⁡2A=sec⁡2A1-\sin^2A=\sec^2A.
D.Use 1−sin⁡2A=cos⁡2A1-\sin^2A=\cos^2A.
Question 7mcq
Simplify (1−sin⁡A)(1+sin⁡A)(1-\sin A)(1+\sin A).
A.1+sin⁡2A1+\sin^2A
B.tan⁡2A\tan^2A
C.cos⁡2A\cos^2A
D.sin⁡2A\sin^2A
Question 8mcq
If tan⁡A=34\tan A=\frac{3}{4}, find sec⁡2A\sec^2A.
A.74\frac{7}{4}
B.2516\frac{25}{16}
C.916\frac{9}{16}
D.1625\frac{16}{25}
Question 9mcq
Which step is invalid when sin⁡A=0\sin A=0?
A.Dividing both sides by sin⁡A\sin A
B.Adding cos⁡2A\cos^2A to both sides
C.Using sin⁡2A+cos⁡2A=1\sin^2A+\cos^2A=1
D.Substituting sin⁡A=0\sin A=0
Question 10mcq
A proof begins by assuming the identity to be proved and then reaches the same identity. What is the issue?
A.It is always the shortest valid proof.
B.It proves the converse only.
C.It is valid because identities are always true.
D.It uses circular reasoning.