Is cos(75)=cos(45+30) ?

The answer to whether cos(75)=cos(45+30) is True

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prove cos(75)=cos(45+30)

prove

cos (7 5^{\circ}) = cos (4 5^{\circ} + 3 0^{\circ})

True

Solution steps

cos (7 5^{\circ}) = cos (4 5^{\circ} + 3 0^{\circ})

Manipulating left side

cos (7 5^{\circ})

Simplify

cos (7 5^{\circ}) : \frac{6 - 2}{4}

= \frac{- 2 + 6}{4}

Manipulating right side

cos (4 5^{\circ} + 3 0^{\circ})

Rewrite using trig identities

= \frac{6 - 2}{4}

We showed that the two sides could take the same form

\Rightarrow True

\frac{1}{1 + cot ^{2} ( x )} = sin^{2} (x)

sec (θ) \cdot sin (θ) = tan (θ)

1 - sin (x) = (sin (\frac{x}{2}) - cos (\frac{x}{2}))^{2}

sin (θ) = cos (\frac{π}{2} - θ)

\frac{cos ( 2 x )}{cos ^{2} ( x )} = 1 - tan^{2} (x)