Prove the following trigonometric identities- 1-1-sin-A-1-1-sin-A-2-sec2-A

LHS = 11+sinA11+sinA + 11–sinA11–sinA We know that cos^2θ+ sin^2θ = 1 sec θ = 1/cos θ LHS = 1/1+sin A+1/1 sin A = 1+sin A+1 sin A/(1+sin A)(1 sin A) = 2/1 sin^2 ...

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Standard X

Mathematics

Trigonometric Identities

Prove the fol...

Question

Prove the following trigonometric identities:

$\frac{1}{1 + s i n A} + \frac{1}{1 - s i n A} = 2 s e c^{2} A$

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Solution

LHS = 11+sinA11+sinA + 11–sinA11–sinA

We know that $c o s^{2} θ + s i n^{2} θ = 1$

$s e c θ = \frac{1}{c o s θ}$

$L H S = \frac{1}{1 + s i n A} + \frac{1}{1 - s i n A}$

$= \frac{1 + s i n A + 1 - s i n A}{(1 + s i n A) (1 - s i n A)}$

$= \frac{2}{1 - s i n^{2} A}$

$= \frac{2}{c o s^{2} A}$

$= 2 s e c^{2} A = R H S$

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Prove the following trigonometric identities: 11+sin A+11−sin A=2 sec2 A

LHS = 11+sinA11+sinA + 11–sinA11–sinA We know that cos2θ+sin2θ=1 sec θ=1cos θ LHS=11+sin A+11−sin A =1+sin A+1−sin A(1+sin A)(1−sin A) =21−sin2 A =2cos2 A =2sec2 A=RHS

Prove the following trigonometric identities:

$\frac{1}{1 + s i n A} + \frac{1}{1 - s i n A} = 2 s e c^{2} A$