Answer:
[tex]\dfrac{117}{125}[/tex]
Step-by-step explanation:
We are given that:
[tex]cos\alpha = \dfrac{7}{25}[/tex]
[tex]sin \beta = \dfrac{4}{5}[/tex]
To find [tex]cos(\alpha - \beta)[/tex].
As per Formula:
[tex]cos(A-B) =cos A cosB+ sinA sinB[/tex]
Here, A is [tex]\alpha[/tex] and B is [tex]\beta[/tex].
So, formula becomes
[tex]cos(\alpha -\beta)=cos\apha cos\beta+sin\alpha sin \beta ..... (1)[/tex]
Using the following identity to calculate [tex]sin\alpha \text{ and } cos \beta[/tex]:
[tex]sin^2 \theta + cos^2 \theta = 1[/tex]
[tex]sin^{2}\alpha+\dfrac {7^{2} }{25^{2} } = 1\\ \Rightarrow sin\alpha = \sqrt{1-\dfrac{49}{625}}\\\Rightarrow sin\alpha = \dfrac{24}{25}[/tex]
Similarly,
[tex]\dfrac {4^{2} }{5^{2} } + cos^{2}\beta = 1\\ \Rightarrow cos\beta = \sqrt{1-\dfrac{16}{25}}\\\Rightarrow cos\beta = \dfrac{3}{5}[/tex]
Putting values in equation (1):
[tex]cos(\alpha -\beta ) = \dfrac{7}{25} \times \dfrac{3}{5} + \dfrac{24}{25} \times \dfrac{4}{5}\\\Rightarrow \dfrac{21+96}{125}\\\Rightarrow \dfrac{117}{125}[/tex]
