Study about Trigonometric Identities problem
Trigonometry means triangular measurements. These can be done using some trig functions and identities. Trigonometric identities can be Pythagorean, reciprocal, quotient, even, odd, half - angle, sum and difference identities. Here we can study some of the example problems using trig identities. In this article we will illustrate about study about trigonometric identities problems.
Study about trigonometric identities problems:
Study about trigonometric identities problems involves in preparing the trig identities problems. Here we will show some examples,
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Example problem 1- Study about trigonometric identities problems
Prove that `cos( Pi / 2 + A)` = - sinA
Solution:
Given:
`cos(Pi / 2 + A) ` = - sinA
we know that cos (A+ B) = cosAcosB - sinAsinB
Expand using `cos( Pi / 2 + A)` above identities,
`cos( Pi / 2 + A)` = cos `Pi / 2 ` cosA - sin `Pi / 2 ` sinA
= 0 `xx ` cosA – 1 `xx ` sinA
= - sinA
`cos( Pi / 2 + A)` = - sinA
Hence proved.
Example problem 2:
Prove that `(sec^(2)A -1) / (sec^(2)A) ` = sin2A
Solution:
Given: (sec2A -1) / sec 2A
Initially solve the left hand side, (sec2A -1) / sec 2A
We know that (sec2A -1) = tan2A, Substitute the values in the given expression,
=`(tan^(2)A) / (sec^(2)A)`
= (`sin^(2)` A / `cos^(2)` A) / sec 2A where tan2A = sin2A / cos2A
= `(sin^(2)A) / (cos^(2)A)` `xx ` `1/ (cos^(2))` where sec 2A = 1 / cos2A
= `(sin^(2) A) / (cos^(2) A)` ` xx` cos2A
= sin2A
(sec2A -1) / sec 2A = sin2A
Hence proved.
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Example problem 3:
Prove that cosA `sqrt(1+ cot^(2)A)` = `sqrt(cosec^(2)A - 1)`
Solution:
Given: cosA `sqrt(1+ cot^(2)A)` = `sqrt(cosec^(2)A - 1)`
Initially solve the left hand side, cosA `sqrt(1+ cot^(2)A)`
We know that 1+ cot2A = cosec2A, Substitute the values in the given expression,
= cosA `sqrt(cosec^(2)A)`
= cosA cosecA where cosecA = `1 / sinA`
= `cosA / sinA`
= cotA where cotA = `sqrt(cosec^(2)A - 1)`
= `sqrt(cosec^(2)A - 1)`
cosA `sqrt(1+ cot^(2)A)` = `sqrt(cosec^(2)A - 1)`
Hence proved.
Practice problem: Study about trigonometric identities problems
Practice problem 1 - Study about trigonometric identities problems
Prove that cos4A - sin4A = 1 - 2sin2A
Practice problem 2 - Study about trigonometric identities problems
Prove that tanA cotA = secA cscA