Prove the identitity- -1-sin-1-sin-tan- -3 MARKS-

Formula: 1 Mark Proof: 2 Marks LHS=√(1 sin θ/1+sin θ)=√((1 sin θ)/(1+sin θ)×(1 sin θ)/(1 sin θ))[Rationalising the denominator] =√((1 sin θ)^2/1 sin^2θ)=√((1 ...

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Standard X

Mathematics

Relation between Trigonometric Ratios

Prove the ide...

Question

Prove the identitity: $\sqrt{\frac{1 - s i n θ}{1 + s i n θ}} = s e c θ - t a n θ$ [3 MARKS]

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Solution

Formula: 1 Mark
Proof: 2 Marks

$L H S = \sqrt{\frac{1 - s i n θ}{1 + s i n θ}} = \sqrt{\frac{(1 - s i n θ)}{(1 + s i n θ)} \times \frac{(1 - s i n θ)}{(1 - s i n θ})}$ [Rationalising the denominator]
$= \sqrt{\frac{(1 - s i n θ)^{2}}{1 - s i n^{2} θ}} = \sqrt{\frac{(1 - s i n θ)^{2}}{c o s^{2} θ}} = \frac{1 - s i n θ}{c o s θ} = \frac{1}{c o s θ} - \frac{s i n θ}{c o s θ} = s e c θ - t a n θ = R H S$

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Prove the identitity: √1−sin θ1+sin θ=sec θ−tan θ [3 MARKS]

Formula: 1 Mark Proof: 2 Marks LHS=√1−sin θ1+sin θ=√(1−sin θ)(1+sin θ)×(1− sin θ)(1−sin θ)[Rationalising the denominator] =√(1−sin θ)21−sin2θ=√(1−sin θ)2cos2θ=1−sin θcos θ=1cos θ−sin θcos θ=sec θ−tan θ=RHS

Prove the identitity: $\sqrt{\frac{1 - s i n θ}{1 + s i n θ}} = s e c θ - t a n θ$ [3 MARKS]