Solveeit Logo
Login

Question

Question: How do you simplify \(\cos 45\sin 65-\cos 65\sin 45\) using the sum and difference, double angle or ...

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

Explanation

Solution

Now the given expression is in the form of sin⁡(A−B)=sin⁡Acos⁡B−cos⁡Asin⁡B\sin \left( A-B \right)=\sin A\cos B-\cos A\sin B. Now we know that the formula for sin⁡(A−B)\sin \left( A-B \right) is nothing but cosAsinB – sinAcosB. Hence we can write the equation in the form of sin⁡(A−B)\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 opposite sidehypotenuse \dfrac{\text{opposite side}}{\text{hypotenuse }} while cos denotes adjacent sidehypotenuse\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⁡2θ+cos⁡2θ=1{{\sin }^{2}}\theta +{{\cos }^{2}}\theta =1 .
Now let us learn the sum and addition of angles formula.
sin⁡(A+B)=sin⁡Acos⁡B+cos⁡Asin⁡B\sin \left( A+B \right)=\sin A\cos B+\cos A\sin B and sin⁡(A−B)=sin⁡Acos⁡B−cos⁡Asin⁡B\sin \left( A-B \right)=\sin A\cos B-\cos A\sin B .
Similarly for cos we have,
cos⁡(A+B)=cos⁡Acos⁡B−sin⁡Asin⁡B\cos \left( A+B \right)=\cos A\cos B-\sin A\sin B and cos⁡(A−B)=cos⁡Acos⁡B+sin⁡Asin⁡B\cos \left( A-B \right)=\cos A\cos B+ \sin A\sin B
Now Double angle formula for sin and cos are,
sin⁡2A=2sin⁡Acos⁡A\sin 2A=2\sin A\cos A and cos⁡(2A)=cos⁡2A−sin⁡2A\cos \left( 2A \right)={{\cos }^{2}}A-{{\sin }^{2}}A .
Now let us check the given expression cos⁡45sin⁡65−cos⁡65sin⁡45\cos 45\sin 65-\cos 65\sin 45 .
The given expression is in the form of sin⁡Acos⁡B−cos⁡Asin⁡B\sin A\cos B-\cos A\sin B where A = 65 and B = 45.
We know that sin⁡(A−B)=sin⁡Acos⁡B−cos⁡Asin⁡B\sin \left( A-B \right)=\sin A\cos B-\cos A\sin B
Hence using this we get,
⇒sin⁡(65−45)=cos⁡45sin⁡65−cos⁡65sin⁡45\Rightarrow \sin \left( 65-45 \right)=\cos 45\sin 65-\cos 65\sin 45
Now simplifying the above equation we get,
sin⁡(20)=cos⁡45sin⁡65−cos⁡65sin⁡45\sin \left( 20 \right)=\cos 45\sin 65-\cos 65\sin 45
Hence the given equation can be written as sin⁡(20)\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 A2\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⁡(90−A)\sin A=\cos \left( 90-A \right) or sin⁡2θ+cos⁡2θ=1{{\sin }^{2}}\theta +{{\cos }^{2}}\theta =1.