Prove that cosec²x(tan²x-sin²x)=tan²x
Starting with the left-hand side:
cosec^2x(tan^2x - sin^2x)
Using the identities:
cosec^2x = 1/sin^2x
tan^2x = sin^2x/cos^2x
We can rewrite the left-hand side as:
1/sin^2x * (sin^2x/cos^2x - sin^2x)
Simplifying:
1/sin^2x * [(sin^2x - sin^2x*cos^2x)/cos^2x]
Expanding the numerator:
1/sin^2x * [sin^2x(1 - cos^2x)/cos^2x]
Using the identity:
1 - cos^2x = sin^2x
1/sin^2x * [sin^2x*sin^2x/cos^2x]
Simplifying:
1/sin^2x * [sin^4x/cos^2x]
Simplifying the fraction:
1/sin^2x * tan^2x
Multiplying fractions:
tan^2x * 1/sin^2x
This simplifies to:
tan^2x/cos^2x
Using the identity tan^2x = sin^2x/cos^2x:
sin^2x/cos^2x
This is equal to the right-hand side, therefore proving that:
cosec^2x(tan^2x - sin^2x) = tan^2x.
To prove that cosec^2(x) * (tan^2(x) - sin^2(x)) = tan^2(x), we will start with the left-hand side and simplify it step by step.
Step 1: Start with the left-hand side of the equation:
cosec^2(x) * (tan^2(x) - sin^2(x))
Step 2: Rewrite cosec^2(x) as 1/sin^2(x):
(1/sin^2(x)) * (tan^2(x) - sin^2(x))
Step 3: Distribute the numerator (1) over both terms inside the parentheses:
(1 * tan^2(x))/sin^2(x) - (1 * sin^2(x))/sin^2(x)
Simplified to:
tan^2(x)/sin^2(x) - sin^2(x)/sin^2(x)
Step 4: Simplify tan^2(x)/sin^2(x) by using the identity tan(x) = sin(x)/cos(x):
(sin^2(x)/cos^2(x)) / sin^2(x) - sin^2(x)/sin^2(x)
Simplified to:
sin^2(x) / (cos^2(x) * sin^2(x)) - sin^2(x)/sin^2(x)
Step 5: Cancel out the common factor sin^2(x) in the denominator:
1/cos^2(x) + 1 - sin^2(x)/sin^2(x)
Step 6: Simplify 1/cos^2(x) to sec^2(x):
sec^2(x) + 1 - sin^2(x)/sin^2(x)
Step 7: Use the identity sin^2(x) + cos^2(x) = 1:
sec^2(x) + 1 - 1
Step 8: Simplify sec^2(x) + 1 - 1:
sec^2(x)
Therefore, we have shown that the left-hand side (cosec^2(x) * (tan^2(x) - sin^2(x))) is equal to the right-hand side (tan^2(x)).