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How do you simplify $\cos 45\sin 65-\cos 65\sin 45$ using the sum and difference, double angle or half angle formula.

Last updated date: 15th Jun 2024
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Hint: Now the given expression is in the form of $\sin \left( A-B \right)=\sin A\cos B-\cos A\sin B$. Now we know that the formula for $\sin \left( A-B \right)$ is nothing but cosAsinB – sinAcosB. Hence we can write the equation in the form of $\sin \left( A-B \right)$ and hence we will have a simplified expression.

Complete step-by-step solution:
Now let us first understand the trigonometric identities for cos and sin.
sin and cos are trigonometric ratios. sin denotes $\dfrac{\text{opposite side}}{\text{hypotenuse }}$ while cos denotes $\dfrac{\text{adjacent side}}{\text{hypotenuse}}$ .
Now all other trigonometric identities can be expressed in the form of sin and cos. Now consider let us understand the identities related to these identities.
If we apply Pythagora's theorem on the ratios we get the identity ${{\sin }^{2}}\theta +{{\cos }^{2}}\theta =1$ .
Now let us learn the sum and addition of angles formula.
$\sin \left( A+B \right)=\sin A\cos B+\cos A\sin B$ and $\sin \left( A-B \right)=\sin A\cos B-\cos A\sin B$ .
Similarly for cos we have,
$\cos \left( A+B \right)=\cos A\cos B-\sin A\sin B$ and $\cos \left( A-B \right)=\cos A\cos B+ \sin A\sin B$
Now Double angle formula for sin and cos are,
$\sin 2A=2\sin A\cos A$ and $\cos \left( 2A \right)={{\cos }^{2}}A-{{\sin }^{2}}A$ .
Now let us check the given expression $\cos 45\sin 65-\cos 65\sin 45$ .
The given expression is in the form of $\sin A\cos B-\cos A\sin B$ where A = 65 and B = 45.
We know that $\sin \left( A-B \right)=\sin A\cos B-\cos A\sin B$
Hence using this we get,
$\Rightarrow \sin \left( 65-45 \right)=\cos 45\sin 65-\cos 65\sin 45$
Now simplifying the above equation we get,
$\sin \left( 20 \right)=\cos 45\sin 65-\cos 65\sin 45$
Hence the given equation can be written as $\sin \left( 20 \right)$.

Note: Now note that the double angle formula can be easily obtained by substituting B = A in the addition of angles formula. Similarly by replacing A by $\dfrac{A}{2}$ in the double angle formula we will get the half angle formula. We can also always convert sin and cos with identity $\sin A=\cos \left( 90-A \right)$ or ${{\sin }^{2}}\theta +{{\cos }^{2}}\theta =1$.