Question

sin75°-sin 15°\cos75°+cos15°

Answer 1

The answer is in the picture.

Answer 2

Answer:

$\frac{sin75-sin15}{cos75+cos15}$

=$\frac{\sqrt{3}}{3}$

Explanation:

Given $\frac{sin75-sin15}{cos75+cos15}$

=$\frac{sin(90-15)-sin15}{cos(90-15)+cos15}$

/* i) sin(90-A) = cosA

ii)cos(90-15) = sin15 */

=$\frac{cos15-sin15}{sin15+cos15}$

Multiply numerator and denominator by (cos15 - sin15), we get

= $\frac{(cos15-sin15)(cos15-sin15)}{(cos15+sin15)(cos15-sin15)}$

= $\frac{(cos15-sin15)^{2}}{cos^{2}15-sin^{2}15}$

/* x²-y² = (x+y)(x-y) */

= $\frac{cos^{2}15+sin^{2}15-2cos15sin15}{cos(2\times15)}$

/* cos²A - sin²A = sin2A */

=$\frac{1-sin(2\times15)}{cos30}$

/* 2sinAcosA = sin2A */

= $\frac{1-sin30}{cos30}$

= $\frac{1-\frac{1}{2}}{\frac{\sqrt{3}}{2}}$

= $\frac{\frac{(2-1)}{2}}{\frac{\sqrt{3}}{2}}$

= $\frac{1}{\sqrt{3}}$

Rationalising the denominator, we get

= $\frac{\sqrt{3}}{\sqrt{3}\times \sqrt{3}}$

= $\frac{\sqrt{3}}{3}$

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