The number of solutions in [0, 2π] of the equation 16sin2⁡x+16cos2⁡x=10 , is

Solution:

we have,

$16^{\sin^{2} x} + 16^{\cos^{2} x} = 10$

$\Rightarrow 16^{\sin^{2} x} + \frac{16}{16^{\sin^{2} x}} = 10$

$\Rightarrow y^{2} - 10 y + 16 = 0,$ where $y = 16^{\sin^{2} x}$

$\begin{array}{l} \Rightarrow y = 2, 8 \\ \Rightarrow 16^{\sin^{2} x} = 2, 8 \\ \Rightarrow 2^{4 \sin^{2} x} = 2^{1}, 2^{3} \\ \Rightarrow 4 \sin^{2} x = 1, 3 \\ \Rightarrow \sin^{2} x = \frac{1}{4}, \frac{3}{4} \end{array}$

$\begin{array}{l} \Rightarrow \sin x = \pm \frac{1}{2}, \pm \frac{\sqrt{3}}{2} \\ \Rightarrow x = \frac{π}{6}, \frac{5 π}{6}, \frac{7 π}{6}, \frac{11 π}{6}, \frac{π}{3}, \frac{2 π}{3}, \frac{4 π}{3}, \frac{5 π}{3} \end{array}$

Thus, there are 8 solutions

The number of solutions in $[0, 2 π]$ of the equation $16^{\sin^{2} x} + 16^{\cos^{2} x} = 10$ , is

Solution:

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