Answer:
Trig Identities Simplified
Kiran Raut
1-cos^2 x) csc x
2. sec x / csc x
3. 1 - sin^2 x / csc^2 x-1
4. sec^2 x(1-sin^2x
The expression "1 - cos^2 x) csc x" can be simplified as follows:
1 - cos^2 x = sin^2 x (using the trigonometric identity sin^2 x + cos^2 x = 1)
So the expression becomes: sin^2 x * csc x
The expression "sec x / csc x" can be simplified as follows:
sec x = 1/cos x (using the trigonometric identity sec x = 1/cos x)
csc x = 1/sin x (using the trigonometric identity csc x = 1/sin x)
So the expression becomes: (1/cos x) / (1/sin x)
To divide by a fraction, we can multiply by its reciprocal, so the expression simplifies to: (1/cos x) * (sin x/1)
The expression "1 - sin^2 x / csc^2 x-1" can be simplified as follows:
csc x = 1/sin x (using the trigonometric identity csc x = 1/sin x)
csc^2 x = (1/sin x)^2 = 1/sin^2 x
So the expression becomes: 1 - sin^2 x / (1/sin^2 x) - 1
To divide by a fraction, we can multiply by its reciprocal, so the expression simplifies to: 1 - sin^2 x * sin^2 x - 1
Now we can simplify further using the trigonometric identity sin^2 x * cos^2 x = sin^2 x (1 - sin^2 x), so the expression becomes: 1 - sin^2 x * (1 - sin^2 x)
The expression "sec^2 x(1-sin^2x)" can be simplified as follows:
sec^2 x = (1/cos x)^2 = 1/cos^2 x (using the trigonometric identity sec x = 1/cos x)
So the expression becomes: 1/cos^2 x * (1 - sin^2 x)
Now we can simplify further using the trigonometric identity 1 - sin^2 x = cos^2 x, so the expression becomes: 1/cos^2 x * cos^2 x
The cos^2 x terms cancel out, leaving us with: 1.