If t a n θ = 3 4 , find the value of 1 - c o s θ 1 + c o s θ .

Given: t a n θ = 3 4 Now, we have to find the value of the expression 1 - c o s θ 1 + c o s θ From the above figure, we get, A = A B 2 + B C 2 = 3 x 2 + 4 x 2 = 5 x ⇒ c o s θ = 4 5 Hence, 1 - c o s θ 1 + c o s θ = 1 - 4 5 1 + 4 5 = 1 9 Therefore, the value of the given expression is 1 9 .

If t a n θ = 3 4 , find the value of 1 - c o s θ 1 + c o s θ .

Given: t a n θ = 3 4 Now, we have to find the value of the expression 1 - c o s θ 1 + c o s θ From the above figure, we get, A = A B 2 + B C 2 = 3 x 2 + 4 x 2 = 5 x ⇒ c o s θ = 4 5 Hence, 1 - c o s θ 1 + c o s θ = 1 - 4 5 1 + 4 5 = 1 9 Therefore, the value of the given expression is 1 9 .

If 3 t a n θ = 3 s i n θ , find the value of s i n 2 θ - c o s 2 θ .

Given: 3 t a n θ = 3 s i n θ Now, we have to find the value of s i n 2 θ - c o s 2 θ . 3 t a n θ = 3 s i n θ ⇒ 3 s i n θ c o s θ = 3 s i n θ ∴ t a n θ = s i n θ c o s θ ⇒ c o s θ = 3 3 = 1 3 Therefore, s i n 2 θ - c o s 2 θ = 1 - c o s 2 θ - c o s 2 θ [ ∵ sin 2 ⁡ θ + cos 2 ⁡ θ = 1 ] = 1 - 2 c o s 2 θ = 1 - 2 × 1 3 2 = 1 3 Hence, the value of the given expression is 1 3 .

If cosec θ = 13 12 , find the value of 2 s i n θ - 3 c o s θ 4 s i n θ - 9 c o s θ .

Given, cosec θ = 13 12 Now, we have to find the value of the expression 2 sin θ - 3 cos θ 4 sin θ - 9 cos θ ⇒ s i n θ = 1 c o s e c θ = 1 13 12 = 12 13 c o s θ = 1 - s i n 2 θ = 1 - 12 13 2 = 5 13 Hence, 2 s i n θ - 3 c o s θ 4 s i n θ - 9 c o s θ = 2 × 12 13 - 3 × 5 13 4 × 12 13 - 9 × 5 13 = 3 Therefore, the value of the given expression is 3 .

If s i n θ + c o s θ = 2 c o s ( 90 ° - θ ) , find c o t θ .

Given: s i n θ + c o s θ = 2 c o s 90 ° - θ Now, we have to find the value of cot ⁡ θ . s i n θ + c o sθ = 2 c o s 90 ° - θ ⇒ cosθ = 2 cos 90 ° - θ - sinθ ⇒ cosθ = 2 sinθ - sinθ ⇒ c o s θ = 2 - 1 s i n θ [ ∵ cos 90 ° - θ = s i n θ ] ⇒ c o s θ s i n θ = 2 - 1 ⇒ c o t θ = 2 - 1 ∵ c o s θ s i n θ = tan ⁡ θ Therefore, the value of c o t θ = 2 - 1