alleryStart learning →Verifying (sec^2 x - 1) / sec^2 x = sin^2 x by splitting the fraction gives which intermediate expression?
Atan^2 x - 1
Bcos^2 x - 1
C1 - cos^2 x
D1 + cos^2 x
Answer & Solution
Correct answer: C. 1 - cos^2 x
1. Split the single fraction into sec^2 x / sec^2 x minus 1 / sec^2 x.
2. The first piece is simply 1.
3. The second piece uses the reciprocal identity 1 / sec^2 x = cos^2 x.
4. So the expression becomes 1 - cos^2 x.
5. Rearranging sin^2 x + cos^2 x = 1 gives 1 - cos^2 x = sin^2 x, which is the target.
6. The option cos^2 x - 1 reverses the subtraction and equals minus sin^2 x, so it carries the wrong sign.
7. The option 1 + cos^2 x adds instead of subtracting and can reach 2, which sin^2 x never does.
_Source: OpenStax Algebra and Trigonometry (CC BY 4.0), Ch 9 "Trigonometric Identities and Equations", section 9.1 Solving Trigonometric Equations with Identities_
Keep practising Trigonometric Identities and Equations
Browse all Trigonometric Identities and Equations questions for B.Sc Mathematics, or explore more Mathematics topics.
Related questions
If a ladder's base is a feet from the wall and its length is 4a feet, how high up the wallSafety rules place a ladder's base 1 foot from the wall for every 4 feet of ladder length.Compared with y = sin t, the graph of y = sin(2t) is which transformation?Solving sin(n t) = c across one turn requires going around the unit circle how many times?Solve cos(2t) = 1/2 on the interval from 0 to 2 pi.Solve 3 cos t + 3 = 2 sin^2 t exactly on the interval from 0 to 2 pi.Solve cos t cos(2t) + sin t sin(2t) = sqrt(3)/2 on the interval from 0 to 2 pi.Solve 2 sin^2 t + sin t = 0 exactly on the interval from 0 to 2 pi.