Advertisements
Advertisements
Question
Prove that sin θ (1 – tan θ) – cos θ (1 – cot θ) = cosec θ – sec θ.
Advertisements
Solution
L.H.S. = sin θ (1 – tan θ) – cos θ (1 – cot θ)
= `sin θ (1 - (sin θ)/(cos θ)) - cos θ (1 - (cos θ)/(sin θ))`
= `sin θ - (sin^2θ)/(cosθ) - cos θ + (cos^2θ)/(sinθ)`
= `sin θ + (cos^2θ)/(sinθ) - (sin^2θ)/(cosθ) - cos θ`
= `(sin^2θ + cos^2θ)/(sinθ) - ((sin^2θ + cos^2θ)/(cosθ))`
= `1/(sinθ) - 1/(cosθ)` ...[∵ sin2θ + cos2θ = 1]
= cosec θ – sec θ
= R.H.S.
∴ sin θ (1 – tan θ) – cos θ (1 – cot θ) = cosec θ – sec θ
APPEARS IN
RELATED QUESTIONS
(1 + tan θ + sec θ) (1 + cot θ − cosec θ) = ______.
Prove the following trigonometric identities.
`(1 - cos theta)/sin theta = sin theta/(1 + cos theta)`
Prove the following trigonometric identities.
`(1 - tan^2 A)/(cot^2 A -1) = tan^2 A`
Prove the following trigonometric identities.
`(1 + cot A + tan A)(sin A - cos A) = sec A/(cosec^2 A) - (cosec A)/sec^2 A = sin A tan A - cos A cot A`
Prove the following identities:
`(cosecA - 1)/(cosecA + 1) = (cosA/(1 + sinA))^2`
Prove that:
(tan A + cot A) (cosec A – sin A) (sec A – cos A) = 1
`1/((1+tan^2 theta)) + 1/((1+ tan^2 theta))`
`((sin A- sin B ))/(( cos A + cos B ))+ (( cos A - cos B ))/(( sinA + sin B ))=0`
If `cos theta = 2/3 , "write the value of" ((sec theta -1))/((sec theta +1))`
Find the value of ` ( sin 50°)/(cos 40°)+ (cosec 40°)/(sec 50°) - 4 cos 50° cosec 40 °`
Prove that:
`(sin^2θ)/(cosθ) + cosθ = secθ`
From the figure find the value of sinθ.
If x = a sec θ cos ϕ, y = b sec θ sin ϕ and z = c tan θ, then\[\frac{x^2}{a^2} + \frac{y^2}{b^2}\]
Prove the following identity :
`sin^4A + cos^4A = 1 - 2sin^2Acos^2A`
Prove that :(sinθ+cosecθ)2+(cosθ+ secθ)2 = 7 + tan2 θ+cot2 θ.
Prove that: (1+cot A - cosecA)(1 + tan A+ secA) =2.
Prove that sec2 (90° - θ) + tan2 (90° - θ) = 1 + 2 cot2 θ.
Prove that cos θ sin (90° - θ) + sin θ cos (90° - θ) = 1.
If a cos θ – b sin θ = c, then prove that (a sin θ + b cos θ) = `± sqrt(a^2 + b^2 - c^2)`
Show that tan 7° × tan 23° × tan 60° × tan 67° × tan 83° = `sqrt(3)`.
