What is the simplified value of (\frac{1}{1+\cos x}+\frac{1}{1-\cos x})?
Correct answer: C
Combining denominators gives (\frac{2}{1-\cos^2 x}). This is (\frac{2}{\sin^2 x}=2\cosec^2 x).
View question detailsMathematics
Trigonometric functions and their properties
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Medium · Level 71 · rational-expression,cosine,cosecant
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- (2\sec^2 x)
- (\cosec^2 x)
- (2\cosec^2 x)
- (1)
Medium · Level 71 · trigonometric identities,secant,tangent,reciprocal expressions,class 11 mathematics
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- \(\sec x-\tan x\)
- \(\sec x+\tan x\)
- \(\tan x-\sec x\)
- \(\frac{1}{\sec x-\tan x}\)
Medium · Level 71 · reciprocal-product,cosecant,cotangent
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- (\cosec x-\cot x)
- (\cot x-\cosec x)
- (\cosec x+\cot x)
- (\frac{1}{\cosec x+\cot x})
Medium · Level 71 · amplitude,sine,transformation
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- (6)
- (4)
- (2)
- (8)
Medium · Level 71 · period,cosine,transformation
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- \(\pi\)
- \(2\pi\)
- \(4\pi\)
- \(\frac{\pi}{2}\)
Medium · Level 71 · trigonometric functions,tangent function,domain of tan,undefined values,class 11 mathematics
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- It is defined for all real \(x\).
- It is undefined at \(x=\frac{\pi}{2}+n\pi\), where \(n\in\mathbb Z\).
- Its range is only \([-1,1]\).
- It is an even function.
Medium · Level 71 · trigonometric functions, odd functions, periodicity, fundamental period, class 11 mathematics
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- \(\sin x\)
- \(\cos x\)
- \(\tan x\)
- \(\sec x\)
Medium · Level 71 · secant,tangent,identity
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- (\frac{21}{20})
- (\frac{20}{21})
- (\frac{29}{20})
- (\frac{5}{2})
Medium · Level 71 · simplification,sine,identity
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- (\cos x)
- (\tan x)
- (\sin x)
- (\sec x)
Medium · Level 71 · range of functions,trigonometric functions,cosine squared,intervals,function transformation
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- \([-4,2]\)
- \([-2,2]\)
- \([0,2]\)
- \([-2,6]\)
Medium · Level 72 · trigonometric-functions,properties,co-function
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- \(\sin \theta-\cos \theta\)
- \(\sin \theta+\cos \theta\)
- \(\cos \theta-\sin \theta\)
- (1)
Medium · Level 72 · trigonometric-functions,tangent,identity
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- (1)
- (3)
- \(\frac{1}{3}\)
- (0)
Medium · Level 72 · trigonometric functions,trigonometric identities,pythagorean identity,class 11 mathematics
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- 0
- 2
- 1
- \(\tan \theta\)
Medium · Level 72 · trigonometric-functions,secant,identity
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- (1)
- (\sin \theta)
- (\cos \theta)
- (0)
Medium · Level 72 · trigonometric-functions,quadrant,pythagorean-identity
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- (\frac{12}{13})
- (-\frac{12}{13})
- (\frac{5}{12})
- (\frac{13}{12})
Medium · Level 72 · trigonometric-functions,quadrant,sine
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- (-\frac{4}{5})
- (\frac{4}{5})
- (\frac{3}{4})
- (-\frac{3}{4})
Medium · Level 72 · trigonometric-functions,tangent,odd-function
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- (\tan \theta)
- (-\tan \theta)
- (\cot \theta)
- (-\cot \theta)
Medium · Level 72 · trigonometric-functions,allied-angles,sine
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- (-\sin \theta)
- (\cos \theta)
- (\sin \theta)
- (-\cos \theta)
Medium · Level 72 · trigonometric-functions,allied-angles,cosine
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- (-\cos \theta)
- (\cos \theta)
- (\sin \theta)
- (-\sin \theta)
Medium · Level 72 · trigonometric-functions,allied-angles,tangent
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- (\tan \theta)
- (-\tan \theta)
- (\cot \theta)
- (1)
What is \(\frac{1}{\sec x+\tan x}\) equal to?
Correct answer: A
Multiply the numerator and denominator by \(\sec x-\tan x\). The denominator becomes \((\sec x+\tan x)(\sec x-\tan x)=\sec^2x-\tan^2x=1\). Hence, \(\frac{1}{\sec x+\tan x}=\sec x-\tan x\). Option C is the negative of the required expression, so it is incorrect. Exam tip: use the identity \(\sec^2x-\tan^2x=1\) for reciprocal expressions of this form.
View question detailsWhat is (\frac{1}{\cosec x-\cot x}) equal to?
Correct answer: C
Since ((\cosec x-\cot x)(\cosec x+\cot x)=1). Hence the required reciprocal is (\cosec x+\cot x).
View question detailsWhat is the amplitude of the function (6-2\sin 4x)?
Correct answer: C
Amplitude equals the absolute value of the coefficient. Here the coefficient of (\sin 4x) is (-2), so the amplitude is (2).
View question detailsWhat is the period of the function \(5\cos(\frac{x}{2})\)?
Correct answer: C
The period of (\cos kx) is \(\frac{2\pi}{k}\). Here \(k=\frac{1}{2}\), so the period is \(4\pi\).
View question detailsWhich of the following statements about the function \(\tan x\) is correct?
Correct answer: B
Since \(\tan x=\frac{\sin x}{\cos x}\), it is undefined wherever \(\cos x=0\), i.e., at \(x=\frac{\pi}{2}+n\pi\). Its range is all real numbers, not \([-1,1]\). Exam tip: for a quotient trig function, first check where its denominator becomes zero.
View question detailsWhich of the following trigonometric functions is odd and has fundamental period \(2\pi\)?
Correct answer: A
Since \(\sin(-x)=-\sin x\), \(\sin x\) is an odd function, and its fundamental period is \(2\pi\). Although \(\tan x\) is also odd, its fundamental period is \(\pi\). In exams, verify each condition separately.
View question detailsIf (\sec x-\tan x=\frac{2}{5}), what is the value of (\tan x)?
Correct answer: A
Since ((\sec x-\tan x)(\sec x+\tan x)=1), (\sec x+\tan x=\frac{5}{2}). Subtracting the two equations gives (\tan x=\frac{21}{20}).
View question detailsWhat is the simplified value of (\frac{\sin x-\sin^3 x}{\cos^2 x})?
Correct answer: C
Write the numerator as (\sin x(1-\sin^2 x)). Since (1-\sin^2 x=\cos^2 x), the value is (\sin x).
View question detailsWhat is the range of the function (2-4\cos^2 x)?
Correct answer: B
Since \(0\leq \cos^2 x\leq 1\), we get \(0\leq 4\cos^2 x\leq 4\). Hence, \(2-4\cos^2 x\) has minimum value \(-2\) when \(\cos^2 x=1\), and maximum value \(2\) when \(\cos^2 x=0\). Therefore, its range is \([-2,2]\). The interval \([0,2]\) is incorrect because the function can also take the value \(-2\). Exam tip: for an expression involving \(\cos^2 x\), first use its range \([0,1]\) and then test the endpoint values.
View question detailsIf \(f(\theta)=\sin \theta+\cos \theta\), then \(f\left(\frac{\pi}{2}-\theta\right)\) is equal to what?
Correct answer: B
Because for complementary angles \( \sin\left(\frac{\pi}{2}-\theta\right)=\cos \theta\) and \( \cos\left(\frac{\pi}{2}-\theta\right)=\sin \theta\). In exams remember co-function formulas.
View question detailsIf \(\tan \theta=3\), what is the value of \(\frac{\sin \theta}{\cos \theta}\)?
Correct answer: B
\( \tan \theta=\frac{\sin \theta}{\cos \theta}\). In exams apply the definition directly.
View question detailsIf \(\sin^2 \theta+\cos^2 \theta=k\), what is the value of (k)?
Correct answer: C
The fundamental trigonometric identity is \(\sin^2\theta+\cos^2\theta=1\). Therefore, \(k=1\). Option 2 may result from incorrectly treating both terms as individually equal to their maximum value; it is not true for every \(\theta\). Exam tip: whenever you see \(\sin^2\theta+\cos^2\theta\), replace it with 1.
View question detailsWhat is the simplified value of (\sec^2 \theta-\tan^2 \theta)?
Correct answer: A
From ( \sec^2 \theta=1+\tan^2 \theta), the difference is (1). In exams rearrange the identity.
View question detailsIf (\theta) is in the first quadrant and (\sin \theta=\frac{5}{13}), what is (\cos \theta)?
Correct answer: A
( \cos^2 \theta=1-\sin^2 \theta=\frac{144}{169}), and in the first quadrant it is positive. In exams do not forget quadrant sign.
View question detailsIf (\theta) is in the second quadrant and (\cos \theta=-\frac{3}{5}), what is (\sin \theta)?
Correct answer: B
( \sin^2 \theta=1-\cos^2 \theta=\frac{16}{25}), and (\sin \theta) is positive in the second quadrant. In exams ASTC rule helps.
View question detailsIf (\tan(-\theta)=m), then (m) equals what?
Correct answer: B
( \tan \theta) is an odd function, so ( \tan(-\theta)=-\tan \theta). In exams focus on sign change.
View question detailsWhat is the value of (\sin(\pi-\theta))?
Correct answer: C
( \pi-\theta) lies in the second quadrant and sine remains positive. In exams remember allied angle formulas.
View question detailsWhat is the value of (\cos(\pi-\theta))?
Correct answer: A
( \pi-\theta) is in the second quadrant where cosine is negative. In exams always apply quadrant sign.
View question detailsWhat is the value of (\tan(\pi-\theta))?
Correct answer: B
( \pi-\theta) is in the second quadrant and tangent is negative. In exams pay special attention to the sign of tangent.
View question detailsQUIZ COMPLETE