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Range of Trigonometric Ratios from 0 to 90 Degrees

If asin∅+b co...

Question

If asinø+bcosø=c then prove that acosø-bsinø=(a²+b²-c²)^-1

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Solution

asinΦ+bcosΦ=c

Squaring both sides

a²sin²Φ + b²cos²Φ+2absinΦcosΦ=c²

a²(1-cos²Φ)+b²(1-sin²Φ)+2absinΦcosΦ=c²

a²-a²cos²Φ+b²-b²sin²Φ+2abcosΦsinΦ=c²

a²+b²-c²=(acosΦ-bsinΦ)²

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