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Question
Prove the following identities:
`(cosecA - 1)/(cosecA + 1) = (cosA/(1 + sinA))^2`
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Solution
L.H.S. = `(cosecA-1)/(cosecA+1)`
= `(cosecA - 1)/(cosecA + 1) xx (cosecA + 1)/(cosecA + 1)`
= `(cosec^2A - 1)/(cosecA + 1)^2`
= `cot^2A/(cosecA + 1)^2`
= `(cos^2A/sin^2A)/(1/sinA + 1)^2`
= `(cosA/(1 + sinA))^2` = R.HS.
RELATED QUESTIONS
Prove the following identities:
`tan A - cot A = (1 - 2cos^2A)/(sin A cos A)`
Write the value of `( 1- sin ^2 theta ) sec^2 theta.`
Find the value of `θ(0^circ < θ < 90^circ)` if :
`cos 63^circ sec(90^circ - θ) = 1`
Prove that `sqrt(2 + tan^2 θ + cot^2 θ) = tan θ + cot θ`.
If cosθ + sinθ = `sqrt2` cosθ, show that cosθ - sinθ = `sqrt2` sinθ.
Choose the correct alternative:
cos 45° = ?
tan2θ – sin2θ = tan2θ × sin2θ. For proof of this complete the activity given below.
Activity:
L.H.S = `square`
= `square (1 - (sin^2theta)/(tan^2theta))`
= `tan^2theta (1 - square/((sin^2theta)/(cos^2theta)))`
= `tan^2theta (1 - (sin^2theta)/1 xx (cos^2theta)/square)`
= `tan^2theta (1 - square)`
= `tan^2theta xx square` .....[1 – cos2θ = sin2θ]
= R.H.S
sin(45° + θ) – cos(45° – θ) is equal to ______.
If a sinθ + b cosθ = c, then prove that a cosθ – b sinθ = `sqrt(a^2 + b^2 - c^2)`.
If tan θ = `x/y`, then cos θ is equal to ______.
