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Q.
The value of k for which $( \cos \, x + \sin \, x )^2 + k \, \sin \, x \, \cos \, x - 1 = 0$ is an identity is
Trigonometric Functions
Solution:
Given $\left(\cos x + \sin x\right)^{2} + k \sin x \cos x -1 = 0 \forall x$
$ \Rightarrow \cos^{2} x + \sin^{2} x + 2 \cos x \sin x + k \sin x \cos x - 1 = 0 \forall x$
$ \Rightarrow \left(k + 2\right) \cos x \sin x = 0 \forall x $
$\Rightarrow k + 2 = 0 \Rightarrow k = -2$