Introduction to quotient identities
In trigonometry, the following are called the quotient ratios. They are:
1. tanx = `sinx/cosx`
2. cotx = `cosx/sinx`
3. secx = `1/cosx`
4. cscx = `1/sinx`
The quotient identities are formed by the two trigonometric ratios sine and cosine functions. These quotient identities are very helpful in proving certain identities in trigonometry. These quotient identities can be manipulated in different form so that based on the need we can make use of them to prove the identities. They also can be used in different application problems to find the heights, distances and angle of elevations and angle depressions etc. Though the above things can be achieved by the two basic trigonometrical ratios sine and cosine functions, but it may take too long time to evaluate the problems involved in it. Hence it is better to know and understand the usage of the quotient trigonometric identities. We can also see some relations among them which will be of very helpful not only in trigonometry but it helps in evaluation of certain problems in calculus as well.
We have the following important identities:
1.(i) 1 + tan^2x = sec^2x
(ii) sec^2x - tan^2x = 1
(iii) tan^2x = sec^2x - 1
2. (i) 1 + cot^2x = csc^2x
(ii) csc^2x - cot^2x = 1
(iii) cot^2x = csc^2x - 1
The above identities are used to check whether the given trigonometric expressions or trigonometric equations are true or not.
Now let us solve few problems on quotient identities.
Example Problems on Quotient Identities
Ex 1: Prove the following identity:
`tanx/(secx ** 1)` +` tanx/(secx+1)` = 2cscx.
Sol: LHS =` tanx/(secx **1) + tanx/(secx+1)`
= tanx `[1/(secx ** 1) + 1/(secx +1)]`
= tanx`[( secx + 1 + secx ** 1)]/[(secx ** 1)( secx + 1)]`
= tanx `[(2 secx)/( sec^2x ** 1)]`
= `(2 tanx secx)/(tan^2x)`
= `[2secx]/tanx`
= 2 cscx = RHS.
2. Prove that (cosecx - sinx) ( secx - cosx) = `1/[tanx+cotx]`
Proof: LHS = (cosecx - sinx) ( secx - cosx)
= `(1/sinx - sinx)(1/cosx - cosx)`
= `(1- sin^2x)/sinx xx (1-cos^2x)/cosx`
= `[cos^2x]/sinx xx [sin^2x]/cosx`
= sinx cosx = `[sinx cosx]/[sin^2x + cos^2x]`
= `1/[[ sin^2x]/[sinx cosx] + [cos^2x]/ [sinx cosx]]`
= `1/[tanx + cotx] ` = RHS.
Hence the proof.
3. Show that tan^2x + cot^2x + 2 = cose2x sec^2x.
Proof: LHS = tan^2x + cot^2x + 2
= tan^2x + cot^2x + 2tanx cotx
= (tanx + cotx)2
= `[sinx/cosx + cosx/sinx]^2`
= `[sin^2x + cos^2 ]/ [ sin^2x cos^2x]`
= `1/[sin^2x] xx 1/[cos^2x] ` = cosec^2x sec^2x = RHS.
Hence the proof.
Practice Problems on Quotient Identities
Prove the following identities:
1. (cosec x - cotx )2 = `[1-cosx]/[1+cosx]`.
2. sec4x - sec^2x = tan4x + tan^2x.
In trigonometry, the following are called the quotient ratios. They are:
1. tanx = `sinx/cosx`
2. cotx = `cosx/sinx`
3. secx = `1/cosx`
4. cscx = `1/sinx`
The quotient identities are formed by the two trigonometric ratios sine and cosine functions. These quotient identities are very helpful in proving certain identities in trigonometry. These quotient identities can be manipulated in different form so that based on the need we can make use of them to prove the identities. They also can be used in different application problems to find the heights, distances and angle of elevations and angle depressions etc. Though the above things can be achieved by the two basic trigonometrical ratios sine and cosine functions, but it may take too long time to evaluate the problems involved in it. Hence it is better to know and understand the usage of the quotient trigonometric identities. We can also see some relations among them which will be of very helpful not only in trigonometry but it helps in evaluation of certain problems in calculus as well.
We have the following important identities:
1.(i) 1 + tan^2x = sec^2x
(ii) sec^2x - tan^2x = 1
(iii) tan^2x = sec^2x - 1
2. (i) 1 + cot^2x = csc^2x
(ii) csc^2x - cot^2x = 1
(iii) cot^2x = csc^2x - 1
The above identities are used to check whether the given trigonometric expressions or trigonometric equations are true or not.
Now let us solve few problems on quotient identities.
Example Problems on Quotient Identities
Ex 1: Prove the following identity:
`tanx/(secx ** 1)` +` tanx/(secx+1)` = 2cscx.
Sol: LHS =` tanx/(secx **1) + tanx/(secx+1)`
= tanx `[1/(secx ** 1) + 1/(secx +1)]`
= tanx`[( secx + 1 + secx ** 1)]/[(secx ** 1)( secx + 1)]`
= tanx `[(2 secx)/( sec^2x ** 1)]`
= `(2 tanx secx)/(tan^2x)`
= `[2secx]/tanx`
= 2 cscx = RHS.
2. Prove that (cosecx - sinx) ( secx - cosx) = `1/[tanx+cotx]`
Proof: LHS = (cosecx - sinx) ( secx - cosx)
= `(1/sinx - sinx)(1/cosx - cosx)`
= `(1- sin^2x)/sinx xx (1-cos^2x)/cosx`
= `[cos^2x]/sinx xx [sin^2x]/cosx`
= sinx cosx = `[sinx cosx]/[sin^2x + cos^2x]`
= `1/[[ sin^2x]/[sinx cosx] + [cos^2x]/ [sinx cosx]]`
= `1/[tanx + cotx] ` = RHS.
Hence the proof.
3. Show that tan^2x + cot^2x + 2 = cose2x sec^2x.
Proof: LHS = tan^2x + cot^2x + 2
= tan^2x + cot^2x + 2tanx cotx
= (tanx + cotx)2
= `[sinx/cosx + cosx/sinx]^2`
= `[sin^2x + cos^2 ]/ [ sin^2x cos^2x]`
= `1/[sin^2x] xx 1/[cos^2x] ` = cosec^2x sec^2x = RHS.
Hence the proof.
Practice Problems on Quotient Identities
Prove the following identities:
1. (cosec x - cotx )2 = `[1-cosx]/[1+cosx]`.
2. sec4x - sec^2x = tan4x + tan^2x.
