What is the general solution for cos(x)=(-3)/(5sin^2(x)) ?

The general solution for cos(x)=(-3)/(5sin^2(x)) is No Solution for x\in\mathbb{R}

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cos(x)=(-3)/(5sin^2(x))

cos (x) = \frac{- 3}{5 sin ^{2} ( x )}

No Solution for x \in R

Solution steps

cos (x) = \frac{- 3}{5 sin ^{2} ( x )}

Subtract

\frac{- 3}{5 sin ^{2} ( x )}

from both sides

cos (x) + \frac{3}{5 sin ^{2} ( x )} = 0

Simplify

cos (x) + \frac{3}{5 sin ^{2} ( x )} : \frac{5 sin ^{2} ( x ) cos ( x ) + 3}{5 sin ^{2} ( x )}

\frac{5 sin ^{2} ( x ) cos ( x ) + 3}{5 sin ^{2} ( x )} = 0

\frac{f ( x )}{g ( x )} = 0 \Rightarrow f (x) = 0

5 sin^{2} (x) cos (x) + 3 = 0

Rewrite using trig identities

3 + (1 - cos^{2} (x)) \cdot 5 cos (x) = 0

Solve by substitution

cos (x) \approx 1.22119 \dots

cos (x) = 1.22119 \dots :

No Solution

Combine all the solutions

No Solution for x \in R

5 cos^{2} (x) - 12 sin^{2} (x) = 13

cos^{2} (x) - 0.5 cos (x) = 0

5 sin (x) cos (x) = 2 cos (x)

sin (x) = cos^{3} (x)

a = \frac{( 1 + sin ^{2} ( x ) )}{( 1 - sin ^{2} ( x ) )}

What is the general solution for cos(x)=(-3)/(5sin^2(x)) ?
The general solution for cos(x)=(-3)/(5sin^2(x)) is No Solution for x\in\mathbb{R}