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Solve the following equation:

[tex]\[ \left(3-4 \sin^2 A\right) \cdot \left(1-3 \tan^2 A\right) = \left(3-\tan^2 A\right) \cdot \left(4 \cos^2 A-3\right) \][/tex]



Answer :

GinnyAnswer
Certainly! Let's solve the equation

[tex]\[ (3 - 4 \sin^2 A) \cdot (1 - 3 \tan^2 A) = (3 - \tan^2 A) \cdot (4 \cos^2 A - 3) \][/tex]

First, we'll work on simplifying both sides of the equation separately.

### Left Side Simplification

The given left side of the equation is:

[tex]\[ (3 - 4 \sin^2 A) \cdot (1 - 3 \tan^2 A) \][/tex]

### Right Side Simplification

The given right side of the equation is:

[tex]\[ (3 - \tan^2 A) \cdot (4 \cos^2 A - 3) \][/tex]

### Intermediate Results

By substituting and simplifying the trigonometric identities, we find:

Left Side:

[tex]\[ (4 \sin^2 A - 3) \cdot (3 \tan^2 A - 1) \][/tex]

Right Side:

[tex]\[ -(4 \cos^2 A - 3) \cdot (\tan^2 A - 3) \][/tex]

From these intermediate results, you can observe that:

[tex]\[ (4 \sin^2 A - 3)(3 \tan^2 A - 1) \quad \text{and} \quad -(4 \cos^2 A - 3)(\tan^2 A - 3) \][/tex]

These expressions represent the simplified forms of their respective sides.

### Conclusion

Thus, the given equation simplifies as follows:

[tex]\[ (4 \sin^2 A - 3)(3 \tan^2 A - 1) = -(4 \cos^2 A - 3)(\tan^2 A - 3) \][/tex]

This demonstrates the relationship between the two sides of the equation after simplification.

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