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9. If

    \[\mathbf{tan}\text{ }\mathbf{\theta }\text{ }=\text{ }\mathbf{a}/\mathbf{b},\]

find the value of

    \[\left( \mathbf{cos}\text{ }\mathbf{\theta }\text{ }+\text{ }\mathbf{sin}\text{ }\mathbf{\theta } \right)/\text{ }\left( \mathbf{cos}\text{ }\mathbf{\theta }\text{ }\text{ }\mathbf{sin}\text{ }\mathbf{\theta } \right)\]

CBSE | Class 10 | Maths | RD Sharma | Trigonometric Ratios
9. If

    \[\mathbf{tan}\text{ }\mathbf{\theta }\text{ }=\text{ }\mathbf{a}/\mathbf{b},\]

find the value of

    \[\left( \mathbf{cos}\text{ }\mathbf{\theta }\text{ }+\text{ }\mathbf{sin}\text{ }\mathbf{\theta } \right)/\text{ }\left( \mathbf{cos}\text{ }\mathbf{\theta }\text{ }\text{ }\mathbf{sin}\text{ }\mathbf{\theta } \right)\]

Solution:

Given,

    \[tan\text{ }\theta \text{ }=\text{ }a/b\]

And, we know by definition that

    \[tan\text{ }\theta \text{ }=\]

opposite side/ adjacent side

Thus, by comparison

Opposite side

    \[=\text{ }a\]

and adjacent side

    \[=\text{ }b\]

To find the hypotenuse, we know that by Pythagoras theorem that

Hypotenuse2 = opposite side2 + adjacent side2

⇒ Hypotenuse

    \[=\text{ }\surd ({{a}^{2}}~+\text{ }{{b}^{2}})\]

So, by definition

    \[sin\text{ }\theta \text{ }=\]

opposite side/ Hypotenuse

    \[sin\text{ }\theta \text{ }=\text{ }a/\text{ }\surd ({{a}^{2}}~+\text{ }{{b}^{2}})\]

And,

    \[cos\text{ }\theta \text{ }=\]

adjacent side/ Hypotenuse

    \[cos\text{ }\theta \text{ }=\text{ }b/\text{ }\surd ({{a}^{2}}~+\text{ }{{b}^{2}})\]

Now,

After substituting for

    \[cos\text{ }\theta \]

and sin

    \[\theta ,\]

we have

Hence Proved.

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