Prove the Following Identity : 1 Sin a + Cos a + 1 Sin a − Cos a = 2 Sin a 1 − 2 Cos 2 a

Prove the following identity :

`1/(sinA + cosA) + 1/(sinA - cosA) = (2sinA)/(1 - 2cos^2A)`

Sum

LHS = `1/(sinA + cosA) + 1/(sinA - cosA)`

= `(sinA - cosA + sinA + cosA)/(sin^2A - cos^2A)`

= `(2sinA)/(1 - cos^2A - cos^2A) = (2sinA)/(1 - 2cos^2A)`

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Chapter 20 Trigonometry
Exercise 21.1 | Q 5.02

`"If "\frac{\cos \alpha }{\cos \beta }=m\text{ and }\frac{\cos \alpha }{\sin \beta }=n " show that " (m^2 + n^2 ) cos^2 β = n^2`

Prove that (cosec A – sin A)(sec A – cos A) sec² A = tan A.

if `a cos^3 theta + 3a cos theta sin^2 theta = m, a sin^3 theta + 3 a cos^2 theta sin theta = n`Prove that `(m + n)^(2/3) + (m - n)^(2/3)`

Prove the following identities:

`1/(1 + cosA) + 1/(1 - cosA) = 2cosec^2A`

`(cot^2 theta ( sec theta - 1))/((1+ sin theta))+ (sec^2 theta(sin theta-1))/((1+ sec theta))=0`

prove that `1/(1 + cos(90^circ - A)) + 1/(1 - cos(90^circ - A)) = 2cosec^2(90^circ - A)`

Prove that `cot^2 "A" [(sec "A" - 1)/(1 + sin "A")] + sec^2 "A" [(sin"A" - 1)/(1 + sec"A")]` = 0

Prove that sin⁴A – cos⁴A = 1 – 2 cos²A.

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

(1 + sin A)(1 – sin A) is equal to ______.