Question

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

• A. 2
• B. 4
• C. 6
• D. 8

Hint:

Logic Tip: A symmetric equation involving both $\sin^2 x$ and $\cos^2 x$ in an interval of length $2\pi$ typically generates multiple solutions per quadrant due to the absolute value nature of square roots. If $\sin^2 x = k^2$ (where $0<k^2<1$), expect exactly 4 solutions in $[0, 2\pi]$.

Answer 1

The Correct Option is D

Concept:
Convert an equation with mixed trigonometric exponents into a quadratic equation by utilizing the Pythagorean identity $\cos^2 x = 1 - \sin^2 x$, and then solve for the variable across the given interval using trigonometric principles.
Step 1: Rewrite the equation using a substitution variable.
Use the identity $\cos^2 x = 1 - \sin^2 x$ to write the equation in terms of a single trigonometric function: $$16^{\sin^2 x} + 16^{1-\sin^2 x} = 10$$ $$16^{\sin^2 x} + \frac{16}{16^{\sin^2 x = 10$$ Let $t = 16^{\sin^2 x}$. The equation becomes: $$t + \frac{16}{t} = 10$$
Step 2: Solve the resulting quadratic equation.
Multiply the entire equation by $t$: $$t^2 - 10t + 16 = 0$$ Factor the quadratic equation: $$(t - 8)(t - 2) = 0$$ So, $t = 2$ or $t = 8$.
Step 3: Solve for x in the interval $[0, 2\pi]$ for t = 2.
If $t = 2$: $$16^{\sin^2 x} = 2 \implies (2^4)^{\sin^2 x} = 2^1 \implies 2^{4\sin^2 x} = 2^1$$ Equating exponents: $$4\sin^2 x = 1 \implies \sin^2 x = \frac{1}{4} \implies \sin x = \pm\frac{1}{2}$$ In the interval $[0, 2\pi]$, $\sin x = 1/2$ yields 2 solutions ($x = \pi/6, 5\pi/6$) and $\sin x = -1/2$ yields 2 solutions ($x = 7\pi/6, 11\pi/6$). This is a total of 4 solutions.
Step 4: Solve for x in the interval $[0, 2\pi]$ for t = 8.
If $t = 8$: $$16^{\sin^2 x} = 8 \implies (2^4)^{\sin^2 x} = 2^3 \implies 2^{4\sin^2 x} = 2^3$$ Equating exponents: $$4\sin^2 x = 3 \implies \sin^2 x = \frac{3}{4} \implies \sin x = \pm\frac{\sqrt{3{2}$$ In the interval $[0, 2\pi]$, $\sin x = \sqrt{3}/2$ yields 2 solutions ($x = \pi/3, 2\pi/3$) and $\sin x = -\sqrt{3}/2$ yields 2 solutions ($x = 4\pi/3, 5\pi/3$). This is another 4 solutions. The total number of solutions is $4 + 4 = 8$.