If sec θ + tan θ = 5, then sin θ = _____.

If sec $\theta$ + tan $\theta$ = 5, then sin $\theta$ = _________.

Show answer & explanation

Correct answer: C

Explanation

The correct answer is option 3 i.e. $\frac{12}{13}$ .

sec θ + tan θ = 5 ....(1)

Now, sec²θ - tan²θ = 1

⇒ (sec θ + tan θ)(sec θ - tan θ) = 1

⇒ 5(sec θ - tan θ) = 1

⇒ sec θ - tan θ = 1/5 .....(2)

Adding equations (1) and (2),

2sec θ = 5 + 1/5 = 26/5

⇒ sec θ = 13/5

⇒ cos θ = 5/13

Subtract equation (2) from equation (1),

⇒ 2tan θ = 5 - 1/5 = 24/5

⇒ tan θ = 12/5

⇒ sin θ = tan θ × cos θ = (12/5)(5/13) = 12/13

Therefore, the value of the expression is 12/13.

SSC CHSL 2023 PYQ — Question 51