Advertisements
Advertisements
Question
Prove the following that:
`tan^3θ/(1 + tan^2θ) + cot^3θ/(1 + cot^2θ)` = secθ cosecθ – 2 sinθ cosθ
Advertisements
Solution
LHS: `((sin^3θ)/(cos^3θ))/((1 + sin^2θ)/(cos^2θ)) + ((cos^3θ)/(sin^3θ))/((1 + cos^2θ)/(sin^2θ))`
= `((sin^3θ)/(cos^3θ))/(((cos^2θ + sin^2θ))/cos^2θ) + ((cos^3θ)/(sin^3θ))/(((sin^2θ + cos^2θ))/sin^2θ)`
= `sin^3θ/cosθ + cos^3θ/sinθ`
= `(sin^4θ + cos^4θ)/(cosθsinθ)`
= `((sin^2θ + cos^2θ)^2 - 2 sin^2θ cos^2θ)/(cosθ sinθ)`
= `(1 - 2 sin^2θ cos^2θ)/(cosθ sinθ)`
= `1/(cos θ sinθ) - (2 sin^2θcos^2θ)/(cosθ sinθ)`
= secθ cosec θ – 2 sinθ cosθ
= RHS
APPEARS IN
RELATED QUESTIONS
Prove the following trigonometric identities:
`(\text{i})\text{ }\frac{\sin \theta }{1-\cos \theta }=\text{cosec}\theta+\cot \theta `
If acosθ – bsinθ = c, prove that asinθ + bcosθ = `\pm \sqrt{a^{2}+b^{2}-c^{2}`
Prove the following trigonometric identities.
`"cosec" theta sqrt(1 - cos^2 theta) = 1`
Prove that:
2 sin2 A + cos4 A = 1 + sin4 A
Prove the following identity :
`(cosA + sinA)^2 + (cosA - sinA)^2 = 2`
Prove the following identity :
`cosecA + cotA = 1/(cosecA - cotA)`
Prove the following identity :
`(tanθ + 1/cosθ)^2 + (tanθ - 1/cosθ)^2 = 2((1 + sin^2θ)/(1 - sin^2θ))`
Choose the correct alternative:
1 + tan2 θ = ?
Prove that `((1 + sin θ - cos θ)/( 1 + sin θ + cos θ))^2 = (1 - cos θ)/(1 + cos θ)`.
If tan θ + cot θ = 2, then tan2θ + cot2θ = ?
