Version Conflict: Formulas that could be put everywhere

Math formula that could be put everywhere 

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اَشْهَدُ اَنْ لَّآ اِلٰهَ اِلَّا اللهُ وَحْدَہٗ لَاشَرِيْكَ لَہٗ وَاَشْهَدُ اَنَّ مُحَمَّدًا عَبْدُهٗ وَرَسُولُہٗ

• sin²θ + cos²θ = 1.
• tan²θ + 1 = sec²θ
• cot²θ + 1 = cosec²θ
• sin 2θ = 2 sin θ cos θ
• cos 2θ = cos²θ – sin²θ
• tan 2θ = 2 tan θ / (1 – tan²θ)
• cot 2θ = (cot²θ – 1) / 2 cot θ
• The "sino + coso" formulas likely refer to trigonometric identities that involve combinations of the sine and cosine functions. There are a few important formulas and identities in trigonometry involving sine (sin) and cosine (cos). Here are the key ones:

1. Sum of Sine and Cosine

This identity combines sine and cosine into one expression:

\sin(x) + \cos(x) = \sqrt{2} \cdot \sin\left(x + \frac{\pi}{4}\right)

\sin(x) + \cos(x) = \sqrt{2} \cdot \cos\left(x - \frac{\pi}{4}\right)

These formulas show that a sum of sine and cosine can be rewritten as a single trigonometric function with a phase shift.

2. Sum of Sines

For the sum of two sine functions, the identity is:

\sin(A) + \sin(B) = 2 \cdot \sin\left(\frac{A + B}{2}\right) \cdot \cos\left(\frac{A - B}{2}\right)

3. Sum of Cosines

For the sum of two cosine functions, the identity is:

\cos(A) + \cos(B) = 2 \cdot \cos\left(\frac{A + B}{2}\right) \cdot \cos\left(\frac{A - B}{2}\right)

4. Product-to-Sum Formulas

These formulas are useful when working with products of sine and cosine functions:

\sin(A) \cdot \sin(B) = \frac{1}{2} \left[\cos(A - B) - \cos(A + B)\right]

\cos(A) \cdot \cos(B) = \frac{1}{2} \left[\cos(A - B) + \cos(A + B)\right] ]

\sin(A) \cdot \cos(B) = \frac{1}{2} \left[\sin(A + B) + \sin(A - B)\right]

5. Pythagorean Identity

The most famous identity involving sine and cosine is the Pythagorean identity:

\sin^2(x) + \cos^2(x) = 1

This identity is fundamental to many trigonometric simplifications.

Summary

The sum and product of sine and cosine functions can be simplified using these identities to create more manageable expressions, especially in calculus or physics problems involving waves or oscillations.