If tan θ = `4/3`, show that `(sintheta + cos theta )=7/5`

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#### Solution

Let us consider a right ΔABC, right angled at B and ∠𝐶 = 𝜃

Now, we know that tan 𝜃 = `(AB)/(BC) = 4/3`

So, if BC = 3k, then AB = 4k, where k is a positive number.

Using Pythagoras theorem, we have:

`AC^2 = AB^2 + BC^2 = (4K)^2 + (3K)^2`

`⟹ AC^2 = 16K^2 + 9K^2 = 25K^2`

⟹ AC = 5k

Finding out the values of sin 𝜃 𝑎𝑛𝑑 cos 𝜃 𝑢𝑠𝑖𝑛𝑔 𝑡ℎ𝑒𝑖𝑟 𝑑𝑒𝑓𝑖𝑛𝑖𝑡𝑖𝑜𝑛𝑠, 𝑤𝑒 ℎ𝑎𝑣𝑒:

Sin 𝜃 = `(AB)/(AC) = (4K)/(5K)=4/5`

`Cos theta= (BC)/(AC) =(3K)/(5K)=3/5`

Substituting these values in the given expression, we get:

`(sin theta + cos theta )=(4/5 +3/5)=(7/5) = RHS`

i.e., LHS = RHS

Hence proved.

Concept: Trigonometric Ratios and Its Reciprocal

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