If-asin-bcos-acosec-bsec-then-prove-that-each-term-is-equal-to-a-2-3-b-2-3-a-2-3-b-2-3-

Question Number 216421 by MATHEMATICSAM last updated on 07/Feb/25

If a \sin θ + b \cos θ = a cosec θ + b \sec θ then

prove that each term is equal to

(a^{\frac{2}{3}} - b^{\frac{2}{3}}) \sqrt{a^{\frac{2}{3}} + b^{\frac{2}{3}}} .

Answered by mr W last updated on 07/Feb/25

a \sin θ + b \cos θ = \frac{a}{\sin θ} + \frac{b}{\cos θ} = k, s a y

a \sin θ + b \cos θ = k \dots (i)

a \cos θ + b \sin θ = k \sin θ \cos θ \dots (i i)

a \cos θ + b \sin θ = (a \sin θ + b \cos θ) \sin θ \cos θ

a \cos θ (1 - \sin^{2} θ) + b \sin θ (1 - \cos^{2} θ) = 0

a \cos^{3} θ + b \sin^{3} θ = 0

\Rightarrow \tan θ = - \frac{a^{\frac{1}{3}}}{b^{\frac{1}{3}}}

\Rightarrow \sin θ = \pm \frac{a^{\frac{1}{3}}}{\sqrt{a^{\frac{2}{3}} + b^{\frac{2}{3}}}}

\Rightarrow \cos θ = \mp \frac{b^{\frac{1}{3}}}{\sqrt{a^{\frac{2}{3}} + b^{\frac{2}{3}}}}

k = \frac{a}{\sin θ} + \frac{b}{\cos θ}

= \pm (a^{\frac{2}{3}} - b^{\frac{2}{3}}) \sqrt{a^{\frac{2}{3}} + b^{\frac{2}{3}}} ✓

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