Advertisements
Advertisements
Question
\[\frac{\tan \theta}{\sec \theta - 1} + \frac{\tan \theta}{\sec \theta + 1}\] is equal to
Options
0
1
-1
2
Advertisements
Solution
The given expression is `cot θ/(cotθ-cot 3θ)+tanθ/(tanθ-tan3θ)`
Simplifying the given expression, we have
`cotθ/(cotθ-cot3θ)+ tanθ/(tanθ-tan3θ)`
= `(cosθ/sin)/(cosθ/sinθ-(cos3θ)/(sin3θ))+(sinθ/cosθ)/(sinθ/sinθ-(sin3θ)/(cos3θ))`
=` (cosθ/sinθ)/((cosθsin 3θ-cos3θsinθ)/(sinθ sin3θ))+ (sin θ/cos θ)/((sinθ cos3θ-sin3θ cosθ)/(cosθ cos3θ))`
=` (cosθ sin3θ)/(cosθ sin3θ-cos3θsinθ)+(sinθ cos3θ)/(sinθ cos3θ-sin3θ sinθ)`
=`(cosθ sin3θ)/(cosθsinθ-cos3θsinθ)+(cos3θ sinθ)/(-1(cosθ sin3θ-cos3θ sinθ))`
`= (cosθ sin3θ)/(cosθ sin3θ-cos3θsinθ)-(cos3θsinθ)/(cosθsin3θ-cos3θsinθ)`
`=(cosθsin3θ-cos3θsinθ)/(cosθsin3θ-cos3θsinθ)`
=1
APPEARS IN
RELATED QUESTIONS
If (secA + tanA)(secB + tanB)(secC + tanC) = (secA – tanA)(secB – tanB)(secC – tanC) prove that each of the side is equal to ±1. We have,
Prove that `(tan^2 theta)/(sec theta - 1)^2 = (1 + cos theta)/(1 - cos theta)`
Prove the following trigonometric identities.
`(1/(sec^2 theta - cos 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)`
Prove the following trigonometric identities.
if cos A + cos2 A = 1, prove that sin2 A + sin4 A = 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
If`( 2 sin theta + 3 cos theta) =2 , " prove that " (3 sin theta - 2 cos theta) = +- 3.`
Write the value of `sin theta cos ( 90° - theta )+ cos theta sin ( 90° - theta )`.
Find the value of sin ` 48° sec 42° + cos 48° cosec 42°`
Prove that:
`"tan A"/(1 + "tan"^2 "A")^2 + "Cot A"/(1 + "Cot"^2 "A")^2 = "sin A cos A"`.
\[\frac{x^2 - 1}{2x}\] is equal to
(cosec θ − sin θ) (sec θ − cos θ) (tan θ + cot θ) is equal to
Prove the following identity :
`sinθ(1 + tanθ) + cosθ(1 +cotθ) = secθ + cosecθ`
Prove the following identity :
`(cosecA - sinA)(secA - cosA)(tanA + cotA) = 1`
If x = r sinA cosB , y = r sinA sinB and z = r cosA , prove that `x^2 + y^2 + z^2 = r^2`
Without using trigonometric table , evaluate :
`(sin47^circ/cos43^circ)^2 - 4cos^2 45^circ + (cos43^circ/sin47^circ)^2`
Prove that `(cos θ)/(1 - sin θ) = (1 + sin θ)/(cos θ)`.
Prove that `((1 - cos^2 θ)/cos θ)((1 - sin^2θ)/(sin θ)) = 1/(tan θ + cot θ)`
Prove the following identities.
`(1 - tan^2theta)/(cot^2 theta - 1)` = tan2 θ
If `sec θ = 41/40`, then find values of sin θ, cot θ, cosec θ.
Show that `(cos^2(45^circ + θ) + cos^2(45^circ - θ))/(tan(60^circ + θ) tan(30^circ - θ)) = 1`
