Question

How do you prove ${\sin ^2}\left( x \right) + {\cos ^2}\left( x \right) = 1$ using other trigonometric identities?

Answer 1

Hint: We can use angle addition formula for cosine: $\cos \left( {\alpha + \beta } \right) = \cos \left( \alpha \right)\cos \left( \beta \right) - \sin \left( \alpha \right)\sin \left( \beta \right)$ to prove the given identity. For this substitute $\alpha = x$ and $\beta = - x$ in the angle addition formula for cosine and use trigonometric identities. Simplifying the equation further we can prove the relationship between the squares of sine and cosine.
Formula used:
i). $\cos \left( {\alpha + \beta } \right) = \cos \left( \alpha \right)\cos \left( \beta \right) - \sin \left( \alpha \right)\sin \left( \beta \right)$
ii). $\sin \left( { - x} \right) = - \sin \left( x \right)$
iii). $\cos \left( { - x} \right) = \cos \left( x \right)$

Complete step-by-step solution:
Typically, the given identity is just proven using the unit circle and the Pythagoras Theorem, simple calculus, or a variety of other methods. Even if one proves it using other methods, it is important to remember that we cannot prove all trig identities only using other trig identities, as it would be relying on circular reasoning, and so even if we do prove it using another trig identity, we must use other methods somewhere along the line. It so happens that ${\sin ^2}\left( x \right) + {\cos ^2}\left( x \right) = 1$ is one of the easier identities to prove using other methods, and so is generally done so.
Still, be all that as it may, let's do a proof using the angle addition formula for cosine:
$\cos \left( {\alpha + \beta } \right) = \cos \left( \alpha \right)\cos \left( \beta \right) - \sin \left( \alpha \right)\sin \left( \beta \right)$
Also, note that sine is an odd function and cosine is an even function, meaning $\sin \left( { - x} \right) = - \sin \left( x \right)$ and $\cos \left( { - x} \right) = \cos \left( x \right)$
Now we proceed to the proof.
Let $\alpha = x$ and $\beta = - x$
$ \Rightarrow \cos \left( {x + \left( { - x} \right)} \right) = \cos \left( x \right)\cos \left( { - x} \right) - \sin \left( x \right)\sin \left( { - x} \right)$
$ \Rightarrow \cos \left( 0 \right) = \cos \left( x \right)\cos \left( x \right) + \sin \left( x \right)\sin \left( x \right)$
$\therefore 1 = {\cos ^2}\left( x \right) + {\sin ^2}\left( x \right)$
Hence, ${\sin ^2}\left( x \right) + {\cos ^2}\left( x \right) = 1$.

Note: In this question, we have used trigonometry identity to prove this question. Trigonometry is a branch of mathematics that helps us to study the relationship between the sides and the angles of a triangle. In practical life, trigonometry is used by cartographers (to make maps). It is also used by the aviation and naval industries. In fact, trigonometry is even used by Astronomers to find the distance between two stars. Hence, it has an important role to play in everyday life. The three most common trigonometric functions are the tangent function, the sine, and the cosine function. In simple terms, they are written as 'sin', 'cos', and 'tan'. Hence, trigonometry is not just a chapter to study, in fact, it is being used in everyday life.