Q.1 Which of the following is the Pythagorean identity?
sin²θ + cos²θ = 1
sinθ + cosθ = 1
tan²θ + 1 = secθ
cot²θ + 1 = cscθ
Explanation - The fundamental Pythagorean identity is sin²θ + cos²θ = 1, which holds for all θ.
Correct answer is: sin²θ + cos²θ = 1
Q.2 The identity 1 + tan²θ = ?
csc²θ
sec²θ
cot²θ
cos²θ
Explanation - From the Pythagorean identity, 1 + tan²θ = sec²θ.
Correct answer is: sec²θ
Q.3 Which of the following is true: cot²θ + 1 = ?
csc²θ
sec²θ
sin²θ
cos²θ
Explanation - The identity is cot²θ + 1 = csc²θ.
Correct answer is: csc²θ
Q.4 Simplify (sinθ / cosθ).
tanθ
cotθ
secθ
cscθ
Explanation - By definition, tanθ = sinθ / cosθ.
Correct answer is: tanθ
Q.5 Simplify (1 + cosθ)(1 - cosθ).
sin²θ
cos²θ
tan²θ
sec²θ
Explanation - (1 + cosθ)(1 - cosθ) = 1 - cos²θ = sin²θ.
Correct answer is: sin²θ
Q.6 Which is the double angle identity for sin2θ?
2sinθcosθ
cos²θ - sin²θ
1 - 2sin²θ
2cos²θ - 1
Explanation - The double angle formula for sine is sin2θ = 2sinθcosθ.
Correct answer is: 2sinθcosθ
Q.7 cos²θ - sin²θ is equal to?
cos2θ
sin2θ
tan2θ
sec2θ
Explanation - cos2θ has multiple forms, one of which is cos²θ - sin²θ.
Correct answer is: cos2θ
Q.8 Simplify (1 + tan²θ) / (1 + cot²θ).
1
tan²θ
cot²θ
sec²θ/csc²θ
Explanation - (1 + tan²θ) = sec²θ, (1 + cot²θ) = csc²θ. Since sec²θ / csc²θ = (1/cos²θ) / (1/sin²θ) = sin²θ / cos²θ = tan²θ, but we must carefully verify. Actually, rewriting directly gives (sec²θ)/(csc²θ) = (1/cos²θ)/(1/sin²θ) = sin²θ/cos²θ = tan²θ. Correction: The answer is tan²θ.
Correct answer is: 1
Q.9 The identity for tan2θ is?
2tanθ / (1 - tan²θ)
2sinθcosθ
(1 - 2sin²θ)
cos²θ - sin²θ
Explanation - The double angle formula for tangent is tan2θ = 2tanθ / (1 - tan²θ).
Correct answer is: 2tanθ / (1 - tan²θ)
Q.10 If sinθ = 3/5, what is cos²θ?
9/25
16/25
12/25
25/25
Explanation - Using sin²θ + cos²θ = 1, cos²θ = 1 - (9/25) = 16/25.
Correct answer is: 16/25
Q.11 Which of the following is true for secθ?
1/cosθ
1/sinθ
cosθ/sinθ
sinθ/cosθ
Explanation - By definition, secθ = 1/cosθ.
Correct answer is: 1/cosθ
Q.12 Which of the following represents sin²θ in terms of cos2θ?
(1 + cos2θ)/2
(1 - cos2θ)/2
2cos²θ
2sinθcosθ
Explanation - The half-angle identity is sin²θ = (1 - cos2θ)/2.
Correct answer is: (1 - cos2θ)/2
Q.13 Which of the following represents cos²θ in terms of cos2θ?
(1 + cos2θ)/2
(1 - cos2θ)/2
1 - sin²θ
2cos²θ - 1
Explanation - The half-angle identity is cos²θ = (1 + cos2θ)/2.
Correct answer is: (1 + cos2θ)/2
Q.14 Simplify (1 - cos2θ)/2.
cos²θ
sin²θ
tan²θ
csc²θ
Explanation - By half-angle identity, (1 - cos2θ)/2 = sin²θ.
Correct answer is: sin²θ
Q.15 Simplify (1 + cos2θ)/2.
sin²θ
cos²θ
tan²θ
cot²θ
Explanation - By half-angle identity, (1 + cos2θ)/2 = cos²θ.
Correct answer is: cos²θ
Q.16 The value of sin²θ + cos²θ is always?
0
1
tan²θ
sec²θ
Explanation - The fundamental identity states sin²θ + cos²θ = 1.
Correct answer is: 1
Q.17 If tanθ = 4/3, then sec²θ = ?
25/9
25/16
16/25
9/25
Explanation - Since 1 + tan²θ = sec²θ, sec²θ = 1 + (16/9) = 25/9.
Correct answer is: 25/9
Q.18 The identity cosAcosB + sinAsinB is equal to?
cos(A - B)
cos(A + B)
sin(A + B)
tan(A - B)
Explanation - By product-to-sum identity, cosAcosB + sinAsinB = cos(A - B).
Correct answer is: cos(A - B)
Q.19 The identity cosAcosB - sinAsinB is equal to?
cos(A + B)
cos(A - B)
sin(A + B)
tan(A + B)
Explanation - By sum-to-product identity, cosAcosB - sinAsinB = cos(A + B).
Correct answer is: cos(A + B)
Q.20 sin(A + B) equals?
sinAcosB + cosAsinB
sinAcosB - cosAsinB
cosAcosB - sinAsinB
cosAcosB + sinAsinB
Explanation - The addition formula for sine is sin(A + B) = sinAcosB + cosAsinB.
Correct answer is: sinAcosB + cosAsinB
Q.21 cos(A + B) equals?
cosAcosB - sinAsinB
cosAcosB + sinAsinB
sinAcosB + cosAsinB
sinAcosB - cosAsinB
Explanation - The addition formula for cosine is cos(A + B) = cosAcosB - sinAsinB.
Correct answer is: cosAcosB - sinAsinB
Q.22 sin(A - B) equals?
sinAcosB - cosAsinB
sinAcosB + cosAsinB
cosAcosB - sinAsinB
cosAcosB + sinAsinB
Explanation - The subtraction formula for sine is sin(A - B) = sinAcosB - cosAsinB.
Correct answer is: sinAcosB - cosAsinB
Q.23 cos(A - B) equals?
cosAcosB + sinAsinB
cosAcosB - sinAsinB
sinAcosB + cosAsinB
sinAcosB - cosAsinB
Explanation - The subtraction formula for cosine is cos(A - B) = cosAcosB + sinAsinB.
Correct answer is: cosAcosB + sinAsinB