Question

iii) cos^2 15° - cos^2 30° + cos^2 45° - cos^2 60° + cos^2 75° = 1/2

please help me solve this​

Answer 1

hope this may help you dear

Answer 2

Step-by-step explanation:

Given:

$cos^2 15° - cos^2 30° + cos^2 45° - cos^2 60° + cos^2 75° = \frac{1}{2}$

To find: Prove the expression.

Solution:

Formula used:

$\bold{sin^2\theta+ cos^2\theta=1}\\\\\bold{cos(90°-\theta)=sin\theta}$

We know the value of cos 30° ,cos 45° and cos 60°.

Step 1: Put the values of known angles

$\implies cos^2 15° - cos^2 30° + cos^2 45° - cos^2 60° + cos^2 75° \\ \\ \implies cos^2 15° - \Big( { \frac{ \sqrt{3} }{2} }\Big)^{2} + \Big( { \frac{ 1 }{ \sqrt{2} } }\Big)^{2} -\Big( { \frac{1 }{2} }\Big)^{2} + cos^2 75° \\ \\ \implies cos^2 15° + cos^2 75° - \frac{3}{4} + \frac{1}{2} - \frac{1}{4}$

Step 2: Apply complementary angle formula.

$\implies cos^2 (90 - 75°) + cos^2 75° - \frac{3}{4} + \frac{1}{2} - \frac{1}{4} \\ \\ \implies sin^275°+ cos^2 75° - \frac{3}{4} + \frac{1}{2} - \frac{1}{4}$

Step 3: Apply identity

$\implies 1 - \frac{3}{4} + \frac{1}{2} - \frac{1}{4}$

Step 4: Take LCM and solve

$\implies \frac{4 - 3 + 2 - 1 }{4} \\ \\ \implies \frac{2}{4} \\ \\ \implies \frac{1}{2} \\ \\ = RHS$

Final answer:

$\bold{cos^2 15° - cos^2 30° + cos^2 45° - cos^2 60° + cos^2 75° = \frac{1}{2}}$

Hence proved.

Hope it helps you.

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Find all the values of x, in between the range [0,2π] for the following equation.

2sin(3x)+3cos(3x)=0.

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