Introduction to Trigonometry Assertion–Reason PYQs Class 10 CBSE (With Answers)

Official CBSE answer · Explanation by Team TutorVertex

Assertion: True.
The adjacent side is shorter than the hypotenuse.
Therefore cos θ < 1.

Reason: True.
The cosine ratio is adjacent side divided by hypotenuse.

Link: R gives both the ratio and the side-length comparison that establish A. Answer: A.

Official CBSE answer · Explanation by Team TutorVertex

Assertion: True.
Using sin² A + cos² A = 1, cos² A = 1 − $\frac{\text{1}}{\text{9}}$.
Thus cos² A = $\frac{\text{8}}{\text{9}}$.
Since A is acute, cos A is positive.
Therefore cos A = $\frac{\text{2√2}}{\text{3}}$.

Reason: True.
R states the identity sin² θ + cos² θ = 1.

Link: Applying R and using the acute-angle condition gives exactly the value in A. Answer: A.

Official CBSE answer · Explanation by Team TutorVertex

Assertion: True.
A + C = 180° − 90° = 90°.
Therefore cos (A + C) = cos 90° = 0.

Reason: True.
The angles of a triangle sum to 180°.
Also cos 90° = 0.

Link: R supplies the two facts used to prove A. Answer: A.

Official CBSE answer · Explanation by Team TutorVertex

Assertion: True.
Using sin² θ + cos² θ = 1, 1 + cos² θ = 1.
Therefore cos θ = 0.

Reason: False as a general identity.
The correct identity is sin² θ + cos² θ = 1.
The printed relation holds at θ = 90°, which satisfies the condition in A.
But at θ = 0°, its left-hand side is −1.
CBSE therefore marks R false when read as an identity.

Link: A follows from the correct identity. R, as printed, is not a general identity. Answer: C.

Official CBSE answer · Explanation by Team TutorVertex

Assertion: False.
By cosec² θ − cot² θ = 1, cosec² θ = 1 + 1 = 2.
Since θ is acute, cosec θ = √2.

Reason: True.
R states the identity cosec² θ − cot² θ = 1.

Link: R gives √2, which contradicts the value 2 in A. Answer: D.

Official CBSE answer · Explanation by Team TutorVertex

Assertion: True.
Using sec² θ − tan² θ = 1, 1 − tan² θ = 1.
Therefore tan θ = 0.

Reason: False as a general identity.
The correct identity is sec² θ − tan² θ = 1.
The printed relation holds at θ = 0°, which satisfies the condition in A.
But at θ = 45°, sec² θ + tan² θ = 2 + 1 = 3.
CBSE therefore marks R false when read as an identity.

Link: A is true, but the sign in the printed Reason is incorrect. Answer: C.

Official CBSE answer · Explanation by Team TutorVertex

Assertion: False.
cos² θ = 1 − $\frac{\text{9}}{\text{25}}$ = $\frac{\text{16}}{\text{25}}$.
Because θ is acute, cos θ is positive.
Therefore cos θ = $\frac{\text{4}}{\text{5}}$.

Reason: True.
R states the identity sin² θ + cos² θ = 1.

Link: R gives the squared cosine; the acute-angle condition selects the positive root. Answer: D.

Official CBSE answer · Explanation by Team TutorVertex

Assertion: True.
By sec² θ = 1 + tan² θ, tan² θ = 9 − 1 = 8.
Since θ is acute, tan θ = 2√2.

Reason: False.
The correct identity is sec² θ = 1 + tan² θ.
For example, at 45°, the printed equation would give 2 = 0.

Link: A uses the correct identity; R has the wrong sign. Answer: C.

Official CBSE answer · Explanation by Team TutorVertex

Assertion: True.
(cosec θ − cot θ)(cosec θ + cot θ)
= cosec² θ − cot² θ
= 1.
Hence the two expressions are reciprocals.

Reason: True.
R states the identity cosec² θ − cot² θ = 1.

Link: R makes the product equal to 1 and therefore explains A. Answer: A.

Official CBSE answer · Explanation by Team TutorVertex

Assertion: True.
The product is cosec² θ − cot² θ = 1.
Therefore the expressions are reciprocals.

Reason: False.
From cosec² θ − cot² θ = 1,
cot² θ − cosec² θ = −1.

Link: A is true, but R reverses the subtraction and gives the wrong value. Answer: C.

Official CBSE answer · Explanation by Team TutorVertex

Assertion: True.
For an acute angle, 0 < cos A < 1.
Thus sec A = $\frac{\text{1}}{\text{cos A}}$ > 1.

Reason: False.
The correct identity is sec² A = 1 + tan² A.
At A = 45°, the printed equation gives 2 = √2, which is false.

Link: A is true; the extra square root makes R false. Answer: C.

Official CBSE answer · Explanation by Team TutorVertex

Assertion: True.
At θ = 45°, 2θ = 90°.
Since cos 90° = 0, tan 90° is undefined.

Reason: True.
sin 90° = 1 and cos 90° = 0.
Therefore they are unequal.

Link: R states that two values differ. To explain why tan 90° is undefined, we need the specific fact cos 90° = 0. R does not explain A. Answer: B.

Official CBSE answer · Explanation by Team TutorVertex

Assertion: True.
Let x = cos θ. In the given range, x > 0.
Then sec θ + cos θ = $\frac{\text{1}}{\text{x}}$ + x ≥ 2.

Reason: True.
(x − 1)² ≥ 0.
Dividing by x > 0 gives x − 2 + $\frac{\text{1}}{\text{x}}$ ≥ 0.
Therefore x + $\frac{\text{1}}{\text{x}}$ ≥ 2.

Link: Substitute x = cos θ in R. This gives precisely the inequality in A. Answer: A.

Official CBSE answer · Explanation by Team TutorVertex

Assertion: True.
From cos A + cos² A = 1, cos A = 1 − cos² A.
Using R, cos A = sin² A.
Substitute into the given equation:
sin² A + (sin² A)² = 1.
Therefore sin² A + sin⁴ A = 1.

Reason: True.
R states the identity sin² A + cos² A = 1.

Link: R lets us replace cos A by sin² A in the given equation, proving A. Answer: A.