Trigonometry ratios practice · Mixed difficulty

Option B is correct because both the assertion and the reason are true statements, but the reason does not explain why a unit circle has a radius of 1 unit.

Option C ($- 1 - \cot \,\alpha$) is correct because it considers the range of $\alpha$ and simplifies the expression accordingly.

Option C ($\cos 1$) is correct because the cosine function is decreasing on the interval [0, 1], and since $\cos x$ varies from -1 to 1, $\cos(\cos x)$ is minimized when $\cos x$ is closest to 1 (in magnitude but negative), which is when $\cos x = 1$ or $\cos x = -1$, so the minimum value is $\cos(1)$.

Option A (5) is correct because by using trigonometric identities and the given value of $\tan 6\theta$, the expression simplifies to 5.

Option C (0) is correct because by substituting $\sin x = \cos^2 x$ and simplifying, the expression reduces to 1 - 1 = 0.

The correct answer is 1 because the given expression simplifies to 1 using trigonometric identities and the condition x + y = z.

Option B (1) is correct because the product of tangents from 1 to 89 degrees simplifies to 1 by pairing $\tan \theta$ with $\tan (90^{\circ} - \theta)$.

Option B (4) is correct because solving the equation derived from the sum of angles in an A.P. equal to the sum of interior angles of a polygon yields n=4.

Option B ($\left( {1,\;4} \right) \cup \left( {4,\;5} \right)$) is correct because it represents the values of 'a' for which the function has no critical points, derived from the condition that the derivative has no real solutions.

Option B (0) is correct because the product includes cos(90°), which is 0, making the entire product zero.

Option D (0) is correct because given $\cos x + \cos^2 x = 1$, we can deduce $\cos x = \sin^2 x$. Substituting this into the expression and simplifying leads to 0.

Option C (3) is correct because by using trigonometric identities and the given value of $\cos A$, the expression $16 \cos ^2(A / 2)-32 \sin (A / 2) \sin (5 A / 2)$ simplifies to 3.

Option B (-1) is correct because substituting the values of the trigonometric functions and simplifying the expression leads to the result -1.

Option A ($\mathrm{\theta },\,\,\phi$) is correct because ${x^2} + {y^2} + {\mathrm{z}^2}$ simplifies to ${r^2}$, which is independent of $\mathrm{\theta }$ and $\phi$.

The correct answer is A ($\frac{1-x^2}{2x}$) because simplifying the given expression using trigonometric identities and the given value of $\tan 25^{\circ} = x$ leads to this result.

Option C (${m^3} - 3m + 2n = 0$) is correct because it represents the relationship derived by cubing the first equation and substituting the second equation.

Option C ($9 \frac{1}{2}$) is correct because pairing $\sin^2 x$ with $\sin^2 (90-x) = \cos^2 x$ allows us to use the identity $\sin^2 x + \cos^2 x = 1$, and accounting for the remaining terms.

Option B ($1+2 \sqrt{2}$) is correct because after simplifying the trigonometric expression and substituting the given value, we obtain $1+2 \sqrt{2}$.

Option A ($\mathrm{4}$) is correct because the maximum value of the expression is 4, which occurs when $\sin^2 x = 1$.

Option B ($8 \cos 2 A$) is correct because the expression simplifies to $8 \cos(2A)$ using trigonometric identities.

The correct answer is D (none of these) because the given expression simplifies to the product of the tangents, not a constant value.

Option B (2) is correct because by expanding the determinant and using the trigonometric identity $\tan A + \tan B + \tan C = \tan A \tan B \tan C$ for a triangle, the determinant simplifies to 2.

Option A (1) is correct because the given condition $\sin x + \sin^2 x = 1$ leads to $\sin x = \cos^2 x$, which simplifies the expression to 1.

Option B ($\frac{\sqrt{10}-10}{12}$) is correct because it is the value of tan α obtained by solving the trigonometric equation and considering the given range for α.

Option C ($\sqrt{3} / 2$) is correct because the expression is a direct application of the cosine double angle formula: $\cos(2\theta) = \frac{{1 - \tan^2(\theta)}}{{1 + \tan^2(\theta)}}$ with $\theta = 15^{\circ}$.

Option A (2) is correct because the expression simplifies to 2 using trigonometric identities.

Option B ($\frac{{2}}{{\left(e^x+e^{-x}\right)}}$) is correct because by using the identity $\sec^2 \theta - \tan^2 \theta = 1$, we can find $\cos \theta$.

Option D (11) is correct because the minute and hour hands coincide 11 times in a 12-hour period.

Option B ($a^2+b^2$) is correct because using trigonometric identities, the given expression simplifies to $a^2 + b^2$.

Option B ($\left(\frac{{x}}{{a}}\right)^{2 / 3}-\left(\frac{{y}}{{b}}\right)^{2 / 3}=1$) is correct because it is derived by expressing $\sec \theta$ and $\tan \theta$ in terms of x, y, a, and b, and then using the identity $\sec^2 \theta - \tan^2 \theta = 1$.