prove that secθ + tanθ = cos/(1-sinθ )
Chapter 6 – Trigonometry – Text Book Solution
Practice Set 6.1| Q 6.8 | Page 132
prove that secθ + tanθ = cos/(1-sinθ )
Solution
We can use the trigonometric identity:
secθ = 1/cosθ tanθ = sinθ/cosθ
Substituting these identities into the left-hand side of the equation, we get:
secθ + tanθ = 1/cosθ + sinθ/cosθ
Combining the two fractions, we get:
secθ + tanθ = (1 + sinθ)/cosθ
To obtain the right-hand side of the equation, we use the identity:
cos²θ = 1 – sin²θ
Rearranging this identity, we get:
cos²θ + sin²θ = 1
Dividing both sides by cos²θ, we get:
1 + tan²θ = sec²θ
Substituting this identity into the right-hand side of the equation, we get:
cos/(1-sinθ) = cosθ/((1-sinθ)cosθ) = 1/(1-sinθ/cosθ)
Using the identity tanθ = sinθ/cosθ, we get:
cos/(1-sinθ) = 1/(1-tanθ)
Substituting this into the previous expression, we get:
cos/(1-sinθ) = 1/(1-tanθ) = 1/(1-(sinθ/cosθ))
Simplifying this expression, we get:
cos/(1-sinθ) = cosθ/(cosθ-sinθ)
Therefore, we have:
secθ + tanθ = (1 + sinθ)/cosθ = cosθ/(cosθ-sinθ)
which is the same as:
secθ + tanθ = cos/(1-sinθ)
Hence, we have proved that secθ + tanθ = cos/(1-sinθ) using trigonometric identities.
Chapter 6 – Trigonometry – Text Book Solution
Practice set 6.1 |Q 6.8| P 132
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