What is the general solution for arctan(0.2x)+arctan(0.0625x)=54 ?

The general solution for arctan(0.2x)+arctan(0.0625x)=54 is x=(sqrt(65.45104…)-5.25)/(0.68819…)

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arctan(0.2x)+arctan(0.0625x)=54

arctan (0.2 x) + arctan (0.0625 x) = 5 4^{\circ}

x = \frac{65.45104 \dots - 5.25}{0.68819 \dots}

Solution steps

arctan (0.2 x) + arctan (0.0625 x) = 5 4^{\circ}

Rewrite using trig identities

arctan (\frac{0.2 x + 0.0625 x}{1 - 0.2 x \cdot 0.0625 x}) = 5 4^{\circ}

Apply trig inverse properties

\frac{0.2 x + 0.0625 x}{1 - 0.2 x \cdot 0.0625 x} = \frac{( 3 10 + 5 2 ) 5 - 5}{20}

Solve

\frac{0.2 x + 0.0625 x}{1 - 0.2 x \cdot 0.0625 x} = \frac{( 3 10 + 5 2 ) 5 - 5}{20} : x = - \frac{5.25 + 65.45104 \dots}{0.68819 \dots}, x = \frac{65.45104 \dots - 5.25}{0.68819 \dots}

x = - \frac{5.25 + 65.45104 \dots}{0.68819 \dots}, x = \frac{65.45104 \dots - 5.25}{0.68819 \dots}

Verify solutions by plugging them into the original equation

x = \frac{65.45104 \dots - 5.25}{0.68819 \dots}

sin^{2} (θ) + cos (θ) = 1

sin (θ) = \frac{- 12}{13}

tan (x) = (\frac{3}{4})

50 = 25 sin (\frac{π}{2} (x + 1)) + 40

4 sin (2 x) = 0, - π \leq x \leq π

What is the general solution for arctan(0.2x)+arctan(0.0625x)=54 ?
The general solution for arctan(0.2x)+arctan(0.0625x)=54 is x=(sqrt(65.45104…)-5.25)/(0.68819…)