(B) Given the equation: $(\cos x + \sin x)^2 + k \sin x \cos x - 1 = 0, \forall x$Expanding the square: $(\cos^2 x + \sin^2 x + 2 \sin x \cos x) + k \

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(B) Given the equation: $(\cos x + \sin x)^2 + k \sin x \cos x - 1 = 0, \forall x$Expanding the square: $(\cos^2 x + \sin^2 x + 2 \sin x \cos x) + k \sin x \cos x - 1 = 0, \forall x$Using the identity $\cos^2 x + \sin^2 x = 1$: $1 + 2 \sin x \cos x + k \sin x \cos x - 1 = 0, \forall x$Simplifying the equation: $(2 + k) \sin x \cos x = 0, \forall x$For this to be an identity (true for all $x$),the coefficient must be zero:$2 + k = 0$$k = -2$

Class 11 Mathematics Trigonometrical Ratios, Functions and Identities Maximum and minimum values of trigonometrical functions, Conditional trigonometrical identities

Medium

The value of $k$,for which $(\cos x + \sin x)^2 + k \sin x \cos x - 1 = 0$ is an identity,is

A
$-1$
B
$-2$
C
$0$
D
$1$

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Trigonometrical Ratios, Functions and Identities

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Maximum and minimum values of trigonometrical functions, Conditional trigonometrical identities

The maximum value of $\cos \alpha_1 \cdot \cos \alpha_2 \cdots \cos \alpha_n$ under the restrictions $0 \le \alpha_1, \alpha_2, \dots, \alpha_n \le \frac{\pi}{2}$ and $\cot \alpha_1 \cdot \cot \alpha_2 \cdots \cot \alpha_n = 1$ is

DifficultIIT 2001

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The minimum value of $27^{\cos 2x} 81^{\sin 2x}$ is

MediumKCET 2009

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The maximum value of $3 \cos \theta + 5 \sin \left( \theta - \frac{\pi}{6} \right)$ for any real value of $\theta$ is

DifficultJEE MAIN 2019

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If $x \in (0, \frac{\pi}{4})$,then the expression $\frac{\cos x}{\sin^2 x(\cos x - \sin x)}$ cannot take which of the following values?

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What is the minimum value of $2^{((x^2 - 3)^3 + 27)}$?

Medium

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The value of $k$,for which $(\cos x + \sin x)^2 + k \sin x \cos x - 1 = 0$ is an identity,is

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The maximum value of $\cos \alpha_1 \cdot \cos \alpha_2 \cdots \cos \alpha_n$ under the restrictions $0 \le \alpha_1, \alpha_2, \dots, \alpha_n \le \frac{\pi}{2}$ and $\cot \alpha_1 \cdot \cot \alpha_2 \cdots \cot \alpha_n = 1$ is

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