If secθ + tanθ = p, show that (p^2−1)/(p^2+1)=sinθ - Mathematics

Advertisements
Advertisements
Sum

If secθ + tanθ = p, show that `(p^{2}-1)/(p^{2}+1)=\sin \theta`

Advertisements

Solution 1

We have,

`=(\sec ^{2}\theta +\tan ^{2}\theta +2\sec \theta \tan\theta -1)/(\sec ^{2}\theta +\tan^{2}\theta +2\sec \theta \tan\theta +1)`

`=\frac{(\sec ^{2}\theta -1)+\tan ^{2}\theta +2\sec \theta \tan\theta }{\sec ^{2}\theta +2\sec \theta \tan \theta +(1+\tan^{2}\theta )`

`=(\tan ^{2}\theta +\tan ^{2}\theta +2\sec \theta \tan\theta )/(\sec ^{2}\theta +2\sec \theta \tan \theta +\sec^{2}\theta )`

`=\frac{2\tan ^{2}\theta +2\tan \theta \sec \theta }{2\sec^{2}\theta +2\sec \theta \tan \theta }`

`=\frac{2\tan \theta (\tan \theta +\sec \theta )}{2\sec \theta (\sec\theta +\tan \theta )}`

`=\frac{\tan \theta }{\sec \theta }=\frac{\sin \theta }{\cos \theta \sec\theta }`

= sinθ = RHS

Solution 2

Sec θ + tan θ = P

⇒ `1/cos θ + sin θ /cos θ  = P`

⇒ `(1 + sin θ)/cos θ = P`

⇒ `(1 + sin θ)^2/cos^2 θ = P^2`,      ....(Squaring both sides)

⇒ `(1 + sin^2 θ + 2 sin θ)/cos^2 θ = p^2`

⇒ `(1 + sin^2 θ + 2 sin θ  + cos^2 θ)/(1 + sin^2 θ + 2 sin θ  - cos^2 θ) = (p^2 + 1)/(p^2 - 1)`   ....(Applying componendo and dividendo]

⇒ `(1 + 1 + 2 sin θ)/(sin^2 θ + sin^2 θ + 2 sin θ) = (p^2 + 1)/(p^2 - 1)`

⇒ `(2( 1 + sin θ))/(2 sin θ( 1 + sin θ)) = (p^2 + 1)/(p^2 - 1)`

⇒ `1/sin θ = (p^2 + 1)/(p^2 - 1)`

Taking reciprocals, we get,

⇒ sin θ = `(p^2 - 1)/(p^2 + 1)`

Hence proved.

  Is there an error in this question or solution?
Chapter 18: Trigonometry - Exercise 2

APPEARS IN

ICSE Class 10 Mathematics
Chapter 18 Trigonometry
Exercise 2 | Q 43

RELATED QUESTIONS

 

If `sec alpha=2/sqrt3`  , then find the value of `(1-cosecalpha)/(1+cosecalpha)` where α is in IV quadrant.

 

Prove that sin6θ + cos6θ = 1 – 3 sin2θ. cos2θ.


Prove that:

sec2θ + cosec2θ = sec2θ x cosec2θ


if `cos theta = 5/13` where `theta` is an acute angle. Find the value of `sin theta`


As observed from the top of an 80 m tall lighthouse, the angles of depression of two ships on the same side of the lighthouse of the horizontal line with its base are 30° and 40° respectively. Find the distance between the two ships. Give your answer correct to the nearest meter.


Prove the following trigonometric identities.

`((1 + tan^2 theta)cot theta)/(cosec^2 theta)   = tan theta`


If cos θ + cot θ = m and cosec θ – cot θ = n, prove that mn = 1


Given that:
(1 + cos α) (1 + cos β) (1 + cos γ) = (1 − cos α) (1 − cos α) (1 − cos β) (1 − cos γ)

Show that one of the values of each member of this equality is sin α sin β sin γ


Prove.
`(1-tanA)^2+(1+tanA)^2=2sec^2A`


Prove.
cot2 A - cos2 A = cos2 A.cot2 A


Prove.
(cosec A + sin A) (cosec A - sin A) = cot2 A + cos2


Prove.
`(1+sinA)/cosA+cosA/(1+sinA)=2secA`


Prove.

`tan^2A-tan^2B=(sin^2A-sin^2B)/(cos^2A * cos^2B)`


`(1 + cot^2 theta ) sin^2 theta =1`


`cos^2 theta + 1/((1+ cot^2 theta )) =1`

     


cosec4θ − cosec2θ = cot4θ + cot2θ


`(1+ tan^2 theta)/(1+ tan^2 theta)= (cos^2 theta - sin^2 theta)`


`(cos  ec^theta + cot theta )/( cos ec theta - cot theta  ) = (cosec theta + cot theta )^2 = 1+2 cot^2 theta + 2cosec theta  cot theta`


`{1/((sec^2 theta- cos^2 theta))+ 1/((cosec^2 theta - sin^2 theta))} ( sin^2 theta cos^2 theta) = (1- sin^2 theta cos ^2 theta)/(2+ sin^2 theta cos^2 theta)`


If `(x/a sin a - y/b cos theta) = 1 and (x/a cos theta + y/b sin theta ) =1, " prove that "(x^2/a^2 + y^2/b^2 ) =2`


Write the value of `(cot^2 theta -  1/(sin^2 theta))`. 


Write the value of`(tan^2 theta  - sec^2 theta)/(cot^2 theta - cosec^2 theta)`


If  `sin theta = 1/2 , " write the value of" ( 3 cot^2 theta + 3).`


Prove that:

`(sin^2θ)/(cosθ) + cosθ = secθ`


Prove that:

`"tan A"/(1 + "tan"^2 "A")^2 + "Cot A"/(1 + "Cot"^2 "A")^2 = "sin A cos A"`.


From the figure find the value of sinθ.


sinθ × cosecθ =?


If sec θ + tan θ = x, write the value of sec θ − tan θ in terms of x.


 Write True' or False' and justify your answer  the following : 

The value of  \[\cos^2 23 - \sin^2 67\]  is positive . 


Prove the following identity :

secA(1 + sinA)(secA - tanA) = 1


Choose the correct alternative:

1 + tan2 θ = ?


Prove that: tan (55° + x) = cot (35° - x)


If tan θ = 2, where θ is an acute angle, find the value of cos θ. 


Evaluate:
`(tan 65°)/(cot 25°)`


Prove that `(tan θ)/(cot(90° - θ)) + (sec (90° - θ) sin (90° - θ))/(cosθ. cosec θ) = 2`.


Prove that `sqrt((1 + cos A)/(1 - cos A)) = (tan A + sin A)/(tan A. sin A)`


Prove the following identities.

`(1 - tan^2theta)/(cot^2 theta - 1)` = tan2 θ


Prove the following identities.

sec6 θ = tan6 θ + 3 tan2 θ sec2 θ + 1


Prove the following identities.

`(cot theta - cos theta)/(cot theta + cos theta) = ("cosec"  theta - 1)/("cosec"  theta + 1)`


Prove the following identities.

`(sin "A" - sin "B")/(cos "A" + cos "B") + (cos "A" - cos "B")/(sin "A" + sin "B")`


Prove the following identities.

`(sin^3"A" + cos^3"A")/(sin"A" + cos"A") + (sin^3"A" - cos^3"A")/(sin"A" - cos"A")` = 2


If `sqrt(3)` sin θ – cos θ = θ, then show that tan 3θ = `(3tan theta - tan^3 theta)/(1 - 3 tan^2 theta)`


If cot θ + tan θ = x and sec θ – cos θ = y, then prove that `(x^2y)^(2/3) – (xy^2)^(2/3)` = 1


tan θ cosec2 θ – tan θ is equal to


If (sin α + cosec α)2 + (cos α + sec α)2 = k + tan2α + cot2α, then the value of k is equal to


Prove that `(tan^2 theta - 1)/(tan^2 theta + 1)` = 1 – 2 cos2θ


Choose the correct alternative:

tan (90 – θ) = ?


Prove that sec2θ + cosec2θ = sec2θ × cosec2θ


If sec θ = `41/40`, then find values of sin θ, cot θ, cosec θ


Prove that `(1 + sintheta)/(1 - sin theta)` = (sec θ + tan θ)2 


Prove that `(1 + sin "B")/"cos B" + "cos B"/(1 + sin "B")` = 2 sec B


Prove that `"cot A"/(1 - tan "A") + "tan A"/(1 - cot"A")` = 1 + tan A + cot A = sec A . cosec A + 1


If 3 sin A + 5 cos A = 5, then show that 5 sin A – 3 cos A = ± 3


Show that tan 7° × tan 23° × tan 60° × tan 67° × tan 83° = `sqrt(3)`


If tan θ – sin2θ = cos2θ, then show that sin2 θ = `1/2`.


Prove the following:

`sintheta/(1 + cos theta) + (1 + cos theta)/sintheta` = 2cosecθ


If `sqrt(3) tan θ` = 1, then find the value of sin2θ – cos2θ.


Let x1, x2, x3 be the solutions of `tan^-1((2x + 1)/(x + 1)) + tan^-1((2x - 1)/(x - 1))` = 2tan–1(x + 1) where x1 < x2 < x3 then 2x1 + x2 + x32 is equal to ______.


If 2 cos θ + sin θ = `1(θ ≠ π/2)`, then 7 cos θ + 6 sin θ is equal to ______.


Let α, β be such that π < α – β < 3π. If sin α + sin β = `-21/65` and cos α + cos β = `-27/65`, then the value of `cos  (α - β)/2` is ______.


Proved that `(1 + secA)/secA = (sin^2A)/(1 - cos A)`.


`1/sin^2θ - 1/cos^2θ-1/tan^2θ-1/cot^2θ-1/sec^2θ-1/("cosec"^2θ) = -3`, then find the value of θ.


Find the value of sin2θ  + cos2θ

Solution:

In Δ ABC, ∠ABC = 90°, ∠C = θ°

AB2 + BC2 = `square`   .....(Pythagoras theorem)

Divide both sides by AC2

`"AB"^2/"AC"^2 + "BC"^2/"AC"^2 = "AC"^2/"AC"^2`

∴ `("AB"^2/"AC"^2) + ("BC"^2/"AC"^2) = 1`

But `"AB"/"AC" = square and "BC"/"AC" = square`

∴ `sin^2 theta  + cos^2 theta = square` 


Share
Notifications



      Forgot password?
Use app×