Question

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Answer 1

Step-by-step explanation:

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Answer 2

Question :

$\sf{Prove\:that\:\dfrac{sin\theta + tan\theta}{cos\theta} = tan\theta(1 + sec\theta)}$

Solution :

$:\implies \sf{Prove\:that\:\dfrac{sin\theta + tan\theta}{cos\theta} = tan\theta(1 + sec\theta)}$

From the above equation, we get :

• $\sf{LHS : \dfrac{sin\theta + tan\theta}{cos\theta}}$

• $\sf{RHS : tan\theta(1 + sec\theta)}$

Method (I) :

By solving the LHS, we get :

$\sf{LHS : \dfrac{sin\theta + tan\theta}{cos\theta}} \\ \\ \sf{= \dfrac{sin\theta}{cos\theta} + \dfrac{tan\theta}{cos\theta}}$

We know that, tan(x) = sin(x)/cos(x), so by substituting it in the equation, we get :

$\sf{= tan\theta + \dfrac{sin\theta}{cos\theta\cdot cos\theta}} \\ \\ \sf{= tan\theta + tan\theta \times \dfrac{1}{cos\theta}}$

Now, by substituting 1/cos(x) = sec(x), we get :

$\sf{= tan\theta + tan\theta \times sec\theta} \\ \\ \sf{= tan\theta(1 + sec\theta)}$

Hence, $\sf{LHS = tan\theta(1 + sec\theta)}$

Thus, $\sf{\dfrac{sin\theta + tan\theta}{cos\theta} = tan\theta(1 + sec\theta)}$

[Proved]

Method (II) :

By solving the RHS, we get :

$\sf{RHS : tan\theta(1 + sec\theta)} \\ \\ \sf{= tan\theta + tan\theta\cdot sec\theta}$

By substituting tan(x) = sin(x)/cos(x) in the equation, we get :

$\sf{= \dfrac{sin\theta}{cos\theta} + \dfrac{sin\theta}{cos\theta}\cdot\dfrac{1}{cos\theta}} \\ \\ \sf{= \dfrac{sin\theta}{cos\theta} + \dfrac{tan\theta}{cos\theta}} \left[\because \sf{\dfrac{sin\theta}{cos\theta} = tan\theta}\right] \\ \\ \sf{= \dfrac{sin\theta + tan\theta}{cos\theta}}$

Hence, $\sf{RHS :\dfrac{sin\theta + tan\theta}{cos\theta}}$

Thus, $\sf{\dfrac{sin\theta + tan\theta}{cos\theta} = tan\theta(1 + sec\theta)}$

[Proved]