Prove cos2 -tan2 - -1-sin2 -tan2 - -4 MARKS-

Concept: 2 Marks Application: 2 Marks cos^2 θ+tan^2 θ 1/sin^2 θ = cos^2 θ+tan^2 θ cos^2 θ sin^2 θ/sin^2 θ =tan^2 θ sin^2 θ/sin^2 θ =sin^2 θ/cos^2 θ sin^2 θ/ ...

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Standard XII

Mathematics

General Solution of Trigonometric Equation

Prove cos2 θ+...

Question

Prove $\frac{c o s^{2} θ + t a n^{2} θ - 1}{s i n^{2} θ} = t a n^{2} θ$ [4 MARKS]

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Solution

Concept: 2 Marks
Application: 2 Marks

$\frac{c o s^{2} θ + t a n^{2} θ - 1}{s i n^{2} θ} = \frac{c o s^{2} θ + t a n^{2} θ - c o s^{2} θ - s i n^{2} θ}{s i n^{2} θ}$

$= \frac{t a n^{2} θ - s i n^{2} θ}{s i n^{2} θ}$

$= \frac{\frac{s i n^{2} θ}{c o s^{2} θ} - s i n^{2} θ}{s i n^{2} θ}$

$= \frac{s i n^{2} θ - s i n^{2} θ c o s^{2} θ}{s i n^{2} θ c o s^{2} θ}$

$= \frac{s i n^{2} θ (1 - c o s^{2} θ)}{s i n^{2} θ c o s^{2} θ}$

$= \frac{s i n^{2} θ}{c o s^{2} θ}$

$= t a n^{2} θ$

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Prove cos2 θ+tan2 θ−1sin2 θ=tan2 θ [4 MARKS]

Concept: 2 Marks Application: 2 Marks cos2 θ+tan2 θ−1sin2 θ=cos2 θ+tan2 θ−cos2 θ−sin2 θsin2 θ =tan2 θ−sin2 θsin2 θ =sin2 θcos2 θ−sin2 θsin2 θ =sin2 θ−sin2 θ cos2 θsin2 θ cos2 θ =sin2 θ(1−cos2 θ)sin2 θ cos2 θ =sin2 θcos2 θ =tan2 θ

Prove $\frac{c o s^{2} θ + t a n^{2} θ - 1}{s i n^{2} θ} = t a n^{2} θ$ [4 MARKS]